---
abstract: 'We present constraints on the dark energy equation-of-state parameter, $w=P/(\rho c^2)$, using [60]{} Type Ia supernovae ([SNe Ia]{}) from the ESSENCE supernova survey. We derive a set of constraints on the nature of the dark energy assuming a flat Universe. By including constraints on ([${\Omega}_{\rm M}$]{}, $w$) from baryon acoustic oscillations, we obtain a value for a static equation-of-state parameter $w=$[$-1.05^{+0.13}_{-0.12}~{\rm (stat}~1\sigma{)} \pm 0.13~{\rm (sys)}$]{} and [${\Omega}_{\rm M}$]{}=[$0.274^{+0.033}_{-0.020}~{\rm (stat}~1\sigma{)}$]{} with a best-fit [$\chi^2/{\rm DoF}$]{} of [$0.96$]{}. These results are consistent with those reported by the SuperNova Legacy Survey in a similar program measuring supernova distances and redshifts. We evaluate sources of systematic error that afflict supernova observations and present Monte Carlo simulations that explore these effects. Currently, the largest systematic currently with the potential to affect our measurements is the treatment of extinction due to dust in the supernova host galaxies. Combining our set of ESSENCE [SNe Ia]{} with the SuperNova Legacy Survey [SNe Ia]{}, we obtain a joint constraint of $w=$[$-1.07^{+0.09}_{-0.09}~{\rm (stat}~1\sigma{)} \pm 0.13~{\rm (sys)}$]{}, [${\Omega}_{\rm M}$]{}=[$0.267^{+0.028}_{-0.018}~{\rm (stat}~1\sigma{)}$]{}with a best-fit [$\chi^2/{\rm DoF}$]{} of [$0.91$]{}. The current [SNe Ia]{} data are fully consistent with a cosmological constant.'
author:
- '[W. M. Wood-Vasey]{}, [G. Miknaitis]{}, [C. W. Stubbs]{}, [S. Jha]{}, [A. G. Riess]{}, [P. M. Garnavich]{}, [R. P. Kirshner]{}, [C. Aguilera]{}, [A. C. Becker]{}, [J. W. Blackman]{}, [S. Blondin]{}, [P. Challis]{}, [A. Clocchiatti]{}, [A. Conley]{}, [R. Covarrubias]{}, [T. M. Davis]{}, [A. V. Filippenko]{}, [R. J. Foley]{}, [A. Garg]{}, [M. Hicken]{}, [K. Krisciunas]{}, [B. Leibundgut]{}, [W. Li]{}, [T. Matheson]{}, [A. Miceli]{}, [G. Narayan]{}, [G. Pignata]{}, [J. L. Prieto]{}, [A. Rest]{}, [M. E. Salvo]{}, [B. P. Schmidt]{}, [R. C. Smith]{}, [J. Sollerman]{}, [J. Spyromilio]{}, [J. L. Tonry]{}, [N. B. Suntzeff]{}, and [A. Zenteno]{}'
bibliography:
- 'apj-jour.bib'
- 'cos\_paper.bib'
title: 'Observational Constraints on the Nature of Dark Energy: First Cosmological Results from the ESSENCE Supernova Survey'
---

Introduction: Supernovae and Cosmology {#sec:introduction}
======================================

We report the analysis of [60]{} Type Ia supernovae ([SNe Ia]{}) discovered in the course of the ESSENCE program (Equation of State: Supernovae trace Cosmic Expansion—an NOAO Survey Program) from 2002 to 2005. The aim of ESSENCE is to measure the history of cosmic expansion over the past 5 billion years with sufficient precision to distinguish whether the dark energy is different from a cosmological constant at the $\sigma_w=\pm0.1$ level. Here we present our first results and show that we are well on our way towards that goal. Our present data are fully consistent with a $w=-1$, flat Universe, and our uncertainty in $w$, the parameter that describes the cosmic equation of state, analyzed in the way we outline here, will shrink below $0.1$ for models of constant $w$ as the ESSENCE program is completed. Other approaches to using the luminosity distances have been suggested to constrain possible cosmological models. We here provide the ESSENCE observations in a convenient form suitable for a testing a variety of models.[^1]

As reported in a companion paper [@miknaitis07], ESSENCE is based on a supernova search carried out with the 4-m Blanco Telescope at the Cerro Tololo Inter-American Observatory (CTIO) with the prime-focus MOSAIC II 64 Megapixel CCD camera. Our search produces densely sampled $R$-band and $I$-band light curves for supernovae in our fields. As described in that paper, we optimized the search to provide the best constraints on $w$, given fixed observing time and the properties of the MOSAICII camera and CTIO 4-m telescope. Spectra from a variety of large telescopes, including Keck, VLT, Gemini, and Magellan, allow us to determine supernova types and redshifts. We have paid particular attention to the central problems of calibration and systematic errors that, when the survey is complete in 2008, will be more important to the final precision of our cosmological inferences than statistical sampling errors for about 200 objects.

This first cosmological report from the ESSENCE survey derives some properties of dark energy from the sample presently in hand, which is still small enough that the statistics of the sample size make a noticeable contribution to the uncertainty in dark-energy properties. But our goal is to set out the systematic uncertainties in a clear way so that these are exposed to view and so that we can concentrate our efforts where they will have the most significant effect. To infer luminosity distances to the ESSENCE supernovae over the redshift interval $0.15$–$0.70$, we employ the relations developed for [SNe Ia]{} at low redshift [@jha06c] among their light-curve shapes, colors, and intrinsic luminosities. The expansion history from $z\approx0.7$ to the present provides leverage to constrain the equation-of-state parameter for the dark energy as described below. In §\[sec:introduction\] we sketch the context of the ESSENCE program. In §\[sec:distances\] we show from a set of simulated light curves that this particular implementation of light-curve analysis is consistent, with the same cosmology emerging from the analysis as was used to construct the samples, and that the statistical uncertainty we ascribe to the inference of the dark-energy properties is also correctly measured. This modeling of our analysis chain gives us confidence that the analysis of the actual data set is reliable and its uncertainty is correctly estimated. Section \[sec:systematics\] delineates the systematic errors we confront, estimates their present size, and indicates some areas where improvement can be achieved. Section \[sec:cosmology\] describes the sample and provides the estimates of dark energy properties using the ESSENCE sample. The conclusions of this work are given in §\[sec:conclusions\].

Context
-------

Supernovae have been central to cosmological measurements from the very beginning of observational cosmology. @shapley19 employed supernovae against the “island universe” hypothesis arguing that objects such as SN 1885A in Andromeda would have $M=-16$ which was “out of the question.” Edwin Hubble [@hubble29b] noted “a mysterious class of exceptional novae which attain luminosities that are respectable fractions of the total luminosities of the systems in which they appear.” These extra-bright novae were dubbed “supernovae” by @baade34 and divided into two classes, based on their spectra, by @minkowski41. Type I supernovae (SNe I) have no hydrogen lines while Type II supernovae (SNe II) show H$\alpha$ and other hydrogen lines.

The high luminosity and observed homogeneity of the first handful of SN I light curves prompted @wilson39 to suggest that they be employed for fundamental cosmological measurements, starting with time dilation of their characteristic rise and fall to distinguish true cosmic expansion from “tired light.” After the [SN Ib]{} subclass was separated from the [SNe Ia]{} [see @filippenko97 for a review] this line of investigation has grown more fruitful as techniques of photometry have improved and as the redshift range over which supernovae have been well observed and confirmed to have standard light-curve shapes and luminosities has increased [@rust74; @leibundgut96; @riess97; @goldhaber01; @riess04b; @foley05; @hook05; @conley06; @blondin06]. Within the uncertainties, the results agree with the predictions of cosmic expansion and provide a fundamental test that the underlying assumption of an expanding universe is correct.

Evidence for the homogeneity of [SNe Ia]{} comes from their small scatter in the Hubble diagram. @kowal68 compiled data for the first well-populated Hubble diagram of SNe I. The 1$\sigma$ scatter about the Hubble line was $0.6$ mag, but Kowal presciently speculated that supernova distances to individual objects might eventually be known to 5-10% and suggested that “\[i\]t may even be possible to determine the second-order term in the redshift-magnitude relation when light curves become available for very distant supernovae.” Precise distances to [SNe Ia]{} enable tests for the linearity of the Hubble law and provide evidence for local deviations from the local Hubble flow, attributed to density inhomogeneities in the local universe [@riess95; @riess97; @zehavi98; @bonacic00; @radburn-smith04; @jha06c]. While [SN Ia]{} cosmology is not dependent on the value of $H_0$, it is sensitive to deviations from a homogeneous Hubble flow and these regional velocity fields may limit our ability to estimate properties of dark energy, as emphasized by @hui06 and by @cooray06. Whether the best strategy is to map the velocity inhomogeneities thoroughly or to skip over them by using a more distant low-redshift sample remains to be demonstrated. We have used a lower limit of redshift $z>0.015$ in constructing our sample of [SNe Ia]{}.

The utility of [SNe Ia]{} as distance indicators results from the demonstration that the intrinsic brightness of each [SN Ia]{} is closely connected to the shape of its light curve. As the sample of well-observed [SNe Ia]{} grew, some distinctly bright and faint objects were found. For example, SN 1991T [@filippenko92a; @phillips92] and SN 1991bg [@filippenko92b; @leibundgut93] were of different luminosity, and their light curves were not the same, either. The possible correlation of the shapes of supernova light curves with their luminosities had been explored by @pskovskii77b. More homogeneous photometry from CCD detectors, more extreme examples from larger samples, and more reliable distance estimators enabled @phillips93 to establish the empirical relation between light-curve shapes and supernova luminosities. The Calán-Tololo sample [@hamuy96] and the CfA sample [@riess99; @jha06a], have been used to improve the methods for using supernova light curves to measure supernova distances. Many variations on Phillips’ idea have been developed, including [${\Delta{m_{15}}}$]{} [@phillips99], MLCS [@riess96; @jha06c], DM15 [@prieto06], stretch [@goldhaber01], CMAGIC [@wang03], and SALT [@guy05].

These methods are capable of achieving the 10% precision for supernova distances that [@kowal68] foresaw 40 years ago. In the ESSENCE analysis, we have used a version of the @jha06c method called MLCS2k2. We have compared it with the results of the SALT [@guy05] light-curve fitter used by the SuperNova Legacy Survey [SNLS; @astier06]. This comparison provides a test: if the two approaches do not agree when applied to the same data they cannot both be correct. As shown in §\[sec:distances\], SALT and this version of MLCS2K2, with our preferred extinction prior, are in excellent accord when applied to the same data. While gratifying, this agreement does not prove they are both correct. Moreover, as described in §\[sec:cosmology\], the cosmological results depend somewhat on the assumptions about SN host-galaxy extinction that are employed. This has been an ongoing problem in supernova cosmology. The work of @lira95 demonstrated the empirical fact that although SN Ia have a range of colors at maximum light, they appear to reach the same intrinsic color about 30–90 days past maximum light, independent of light curve shape.

@riess96 used de-reddened [SN Ia]{} data to show that near maximum light intrinsic color differences existed with fainter [SNe Ia]{}appearing redder than brighter objects and then used this information to construct an absorption-free Hubble diagram. Given a good set of observations in several bands, the reddening for individual supernovae can then be determined and the general relations between supernova luminosity and the light curve shapes in many bands can be established [@hamuy96; @riess99; @phillips99]. The initial detections of cosmic acceleration employed either these individual absorption corrections [@riess98] or a full-sample statistical absorption correction [@perlmutter99]. Finding the best approach to this problem, whether by shifting observations to the infrared, limiting the sample to low-extinction cases, or making other restrictive cuts on the data, is an important area for future work. Some ways to explore this issue are sketched in §\[sec:cosmology\].

@kowal68 recognized that second-order terms in cosmic expansion might be measured with supernovae once the precision and redshift range grew sufficiently large. More direct approaches with the [*Hubble Space Telescope (HST)*]{} were imagined by @colgate79 and with special clarity by @tammann79. Tammann anticipated that [*HST*]{} photometry of [SNe Ia]{} at $z\approx0.5$ would lead to a direct determination of cosmic deceleration and that the time dilation of [SN Ia]{} light curves would be a fundamental test of the expansion hypothesis. While [*HST*]{} languished on the ground after the Challenger disaster, this line of research was attempted from the ground at the European Southern Observatory (ESO) by a Danish group in 1986–1988. Their cyclic CCD imaging of the search fields used image registration, convolution and subtraction, and real-time data analysis [@hansen87]. Alas, the rate of [SNe Ia]{} in their fields was lower than they had anticipated, and only one [SN Ia]{}, SN 1988U was discovered and monitored in two years of effort [@hansen87; @norgaard-nielsen89]. More effective searches by the Lawrence Berkeley Lab (LBL) group exploiting larger CCD detectors and sophisticated detection software showed that this approach could be made practical and used to find significant numbers of high-redshift [SNe Ia]{} [@perlmutter95].

By 1995, two groups, the LBL-based Supernova Cosmology Project (SCP) and the High-Z Supernova Search Team [HZT; @schmidt98]) were working in this field. The first [SN Ia]{} cosmology results using 7 high-redshift [SNe Ia]{}[@perlmutter97] found a Universe consistent with [${\Omega}_{\rm M}$]{}$=1$ but subsequent work by the SCP [@perlmutter98] and by the HZT [@garnavich98] revised this initial finding to favor a lower value of [${\Omega}_{\rm M}$]{}. At the January 1998 meeting of the American Astronomical Society both teams reported that the [SN Ia]{} results favored a universe that would expand without limit, but at that time neither team claimed the Universe was accelerating. The subsequent publication of stronger results based on larger samples by the HZT [@riess98] and by the SCP [@perlmutter99] provided a big surprise. The supernova data showed that [SNe Ia]{} at $z\approx0.5$ were about $0.2$ mag dimmer than expected in an open universe and pointed firmly at an accelerating universe [for first-hand accounts, see @overbye99; @riess00b; @filippenko01b; @kirshner02; @perlmutter03]; reviews are given by @leibundgut01, @filippenko05a and others.

The supernova route to cosmological understanding continues to improve. One source of uncertainty has been the small sample of very well observed low-redshift supernovae [@hamuy96; @riess99]. The most recent contribution is the summary of CfA data obtained in 1997–2001 [@jha06a], but significantly enhanced samples from the CfA [@hicken06] together with new data from the Katzman Automatic Imaging Telescope [KAIT; @li00; @filippenko01a; @filippenko05b], from the Carnegie SN Program [@hamuy06], from the Supernova Factory [@wood-vasey04; @copin06], and from the Sloan Digital Sky Survey II Supernova Survey [SDSS II; @frieman04; @dilday05] are in prospect. As the low-$z$ sample approaches 200 objects, the size of the sample will cease to be a source of statistical uncertainty for the determination of cosmological parameters. As described in §\[sec:systematics\], systematic errors of calibration and K-correction will ultimately impose the limits to understanding dark energy’s properties, and we are actively working to improve these areas [@stubbs06].

Some of the potential sources of systematic error in the high-$z$ sample have been examined. The fundamental assumption is that distant [SNe Ia]{} can be analyzed using the methods developed for the low-$z$ sample. Since nearby samples show that the [SNe Ia]{} in elliptical galaxies have a different distribution in luminosity than the [SNe Ia]{} in spirals [@hamuy00; @gallagher05; @neill06; @sullivan06b], morphological classification of the distant sample may provide some useful clues to help improve the cosmological inferences [@williams03]. For example, @sullivan03 showed that restricting the SCP sample to [SNe Ia]{} in elliptical galaxies gave identical cosmological results to the complete sample, which is principally in spiral galaxies. The possibility of grey dust raised by @aguirre99a [@aguirre99b] was examined by @riess00a and and by @nobili05 through infrared observations of high-$z$ supernovae and was put to rest by the very high-redshift observations of @riess04b. Improved methods for handling the vexing problems of absorption by dust have been developed by @knop03 and by @jha06c. These questions are described in more detail in §\[sec:extinction\].

The question of whether distant supernovae have spectra that are the same as nearby supernovae has been investigated by @coil00, @lidman05, @matheson05, @hook05, @howell05, and @blondin06. The more telling question of whether these spectra evolve in the same way as those of nearby objects was approached by @foley05. In all cases, the evidence points toward nearby supernovae behaving in the same way as distant ones, bolstering confidence in the initial results. This observed consistency does not mean that the samples are identical, only that the variations between the nearby and distant samples are successfully accounted for by the methods currently in hand. We do not know whether this will continue to be the case as future investigations press for more stringent limits on cosmological parameters [@albrecht06].

The highest redshift [SN Ia]{} data [@riess04b] show the qualitative signature expected from a mixed dark-energy/dark-matter cosmology. That is, they show cosmic deceleration due to dark matter preceded the current era of cosmic acceleration due to dark energy. The sign of the observed effect on supernova apparent magnitudes reverses—[SNe Ia]{} at $z\sim 0.5$ appear $0.2$ mag dimmer than expected in a coasting cosmology but the very distant supernovae whose light comes from $z>1$ appear brighter than they would in that cosmology. By itself, this turnover is a very encouraging sign that supernova cosmology does not founder on grey dust or even on a simple evolution of supernova properties with cosmic epoch. As part of this analysis, @riess04b constructed the “gold” sample of high-$z$ and low-$z$ supernovae whose observations met reasonable criteria for inclusion in an analysis of all of the published light curves and spectra using a uniform method of deriving distances from the light curves.

The analysis of the gold sample provided an estimate of the time derivative of the equation-of-state parameter, $w$, for dark energy. These observations are very important conceptually because the simplest fact about the cosmological constant as a candidate for dark energy is that it should be constant (i.e., $w' = dw/dz = 0$). The observations are consistent with a constant dark energy over the redshift range out to $z\approx 1.6$. Other forms of dark energy could satisfy the observed constraints, but this observational test is one that the cosmological constant could have failed. In the analysis of the ESSENCE data presented in §\[sec:cosmology\], we use the supernova data to constrain the properties of $w$, as first carried out by @white98 and by @garnavich98. This parameterization of dark energy by $w$ is not the only possible approach. A more detailed approach is to compare the observational data to a specific model and, for example, try to reconstruct the dark energy scalar-field potential [see, for example, @li06]. A more agnostic view is that we are simply measuring the expansion history of the universe, and a kinematic description of that history in terms of expansion rate, acceleration, and jerk [@riess04b; @rapetti06] covers the facts without assuming the nature of dark energy.

The ESSENCE project was conceived to tighten the constraints on dark energy at $z\approx0.5$ to reveal any discrepancy between the observations and the leading candidate for dark energy, the cosmological constant. A simple way to express this is that we aim for a 10% uncertainty in the value of $w$. This program is similar to the approach of the SuperNova Legacy Survey (SNLS) being carried out at the Canada-France-Hawaii Telescope, and we compare our methods and results to theirs [@guy05; @astier06] at several points in the analysis below.

The SNLS has taken the admirable step of publishing their light curves online and making the code of their light-curve fitting program, SALT, available for public inspection and use[^2]. Making the light curves public, as was done for the results of the HZT and its successors @riess98 [@tonry03; @barris04; @krisciunas05; @clocchiatti06], by @knop03, by @riess04b for the very high redshift [*HST*]{} supernova program, and for the low-$z$ data of @hamuy96, @riess99, and @jha06a, provides the opportunity for others to perform their own analysis of the results. In addition to exploring a variety of approaches to analyzing our own [SN Ia]{} observations, we show the first joint constraints from ESSENCE and SNLS, and some joint constraints derived from combining these with the @riess04b gold sample in §\[sec:cosmology\].

Luminosity Distance Determination {#sec:distances}
=================================

The physical quantities of interest in our cosmological measurements are the redshifts and distances to a set of space-time points in the Universe. The redshifts come from spectra and the luminosity distances, $D_L$, come from the observed flux of the supernova combined with our understanding of [SN Ia]{} light curves from nearby objects.

Extracting a luminosity distance to a supernova from observations of its light curve necessitates a number of assumptions. We use the observations of nearby supernovae to establish the relations between color, light-curve shape in multiple bands, and peak luminosity. These nearby observations attain high signal-to-noise ratios, and the nearby objects can be observed in more passbands (including infrared) than faint distant objects. We assume that the resulting method of converting light curves to luminosity distances applies at all redshifts. The observed spectral uniformity of supernovae over a range of redshift [@lidman05; @hook05; @blondin06] supports this approach. We assume that $R_V$, the ratio of selective to absolute extinction, is independent of redshift. Below in §\[sec:extinction\], we test the potential systematic effect of departures from this assumption. We adopt an astrophysically sensible prior distribution of host-galaxy extinction properties, with a redshift dependence that is derived from the simulations we present below.

Our approach is to conduct comprehensive simulations of the ESSENCE data and analysis. As described by @miknaitis07, we use this same approach to explore our photometric performance. For the aspects of our analysis that are “downstream” of the light-curve generation, we generate sets of synthetic light curves and subject them to our analysis pipeline. In this way we can test the performance of our distance-fitting tools, and by exaggerating various systematic errors (zeropoint offsets, etc.) we can assess the impact of these effects on our determination of $w$.

We must recognize and emphasize that in the era of precision [SN Ia]{}cosmology (constraining dark energy properties, rather than just detecting its existence), careful attention to systematic errors is of paramount importance: shifts of a few hundredths of a magnitude can lead to constraints on $w$ that change by $0.1$. Different, yet defensible, choices in the analysis chain may show such effects.

Extracting Luminosity Distances from Light Curves: Distance Fitters
-------------------------------------------------------------------

We use the MLCS2k2 method of @jha06c as the primary tool to derive relative luminosity distances to our [SNe Ia]{}. For comparison, we also provide the results obtained using the Spectral Adaptive Lightcurve Template (SALT) fitter of @guy05 on the ESSENCE light curves. SALT was used in the recent cosmological results paper from the SNLS [@astier06, hereafter A06]. We provide a consistent and comprehensive set of distances obtained to nearby, ESSENCE, and SNLS supernovae for each luminosity-distance fitting technique. The ESSENCE light curves used in this analysis were presented by @miknaitis07 and we provide them online, together with our set of previously published light curves for nearby [SNe Ia]{}, for the convenience of those interested.[^3] Additional [SN Ia]{} light-curve fitting methods will be further explored in future ESSENCE analyses. Understanding the behavior of our distance determination method is critical to our goal of quantifying the uncertainties of our analysis chain.

MLCS2k2 and SALT, as well as the light-curve “stretch” approach used by @perlmutter97 [@perlmutter99], @goldhaber01 and @knop03, exploit the fact that the rate of decline, the color, and the intrinsic luminosity of [SNe Ia]{}are correlated. At present we treat [SNe Ia]{} as a single-parameter family, and the distance fitting techniques use multi-color light curves to deduce a luminosity distance and host-galaxy reddening for each supernova. Previous papers have shown that the different techniques produce relative luminosity distances that scatter by $\sim 0.10$ mag for an individual [SN Ia]{} [e.g., @tonry03], but this scatter is uncorrelated with redshift. As a consequence, the cosmology results are insensitive to the distance fitting technique. However, as described by @miknaitis07, the measurement of the equation-of-state parameter hinges on subtle distortions in the Hubble diagram, so we have undertaken a comprehensive set of simulations to understand potential biases introduced by MLCS2k2.

The MLCS2k2 approach [@riess96; @riess98; @jha06c] to determining luminosity distances uses well-observed nearby [SNe Ia]{} to establish a set of light-curve templates in multiple passbands. The parameters $\Delta$ (roughly equivalent to the variation in peak visual luminosity, this parameter characterizes intrinsic color, rate of decline, and peak brightness), $A_V$ (the $V$-band extinction of the supernova light in its host galaxy), and $\mu$ (the distance modulus) are then determined by fitting each multi-band set of distant supernova light curves to redshifted versions of these templates. Jha et al. (2006c) present results from MLCS2k2 based on nearby SN Ia. Here we have modified MLCS2k2 for application to both high and low-redshift [SNe Ia]{}. We begin with a rest-frame model of the [SN Ia]{} in its host galaxy, and then propagate the model light curves through the host-galaxy extinction, K-correction, Milky Way extinction to the detector, incorporating the measured passband response (including the atmosphere for ground-based observations). We then fit this model directly to the natural-system observations. This forward-modeling approach has particular advantages in application to the more sparsely sampled (in color and time) data typical of high-redshift SN searches.

The SALT method of @guy05, which was used for the SNLS first-results analysis of A06, constructs a fiducial [SN Ia]{} template using combined spectral and photometric information, then transforms this template into the rest frame of the [SN Ia]{}, and finally calculates a flux, stretch, and generalized color. The color parameter in SALT is notable in that it includes both the intrinsic variation in [SN Ia]{} color and the extinction from dust in the host galaxy within a single parameter (in contrast, MLCS2k2 attempts to separate these components of the observed colors for each supernova). While the reddening vector (attenuation vs. color excess) is similar to the [SN Ia]{} color vs. absolute magnitude relation, the two sources of correlated color and luminosity variation are not identical.

The stretch and color parameters of SALT were used by A06 to estimate luminosity distances by fitting for the stretch-luminosity and color-luminosity relationships in the nearby sample and applying those to the full [SNe Ia]{} sample. Given that the SALT color parameter conflates the two physically distinct phenomena of host-galaxy extinction and [SN Ia]{} color variation, it is remarkable and perhaps a source of deep insight that this treatment works as well as it does. Because of both survey selection effects and possible demographic shifts in the host environments of [SNe Ia]{}we would not expect that the proportion of reddening from dust and from intrinsic variation would remain constant with redshift as this approach assumes. However, the SALT/A06 method does seem to work quite well in practice.

Sensitivity to Assumptions about the Host-Galaxy Extinction Distribution: Extinction Priors {#sec:extinction_prior}
-------------------------------------------------------------------------------------------

The best way to treat host-galaxy extinction is a serious question for this work and for the field of supernova cosmology. The Bayesian approach we use is detailed in §\[sec:priors\]. Here we describe simulations that are designed to evaluate the effects of those methods.

There have been four basic approaches to combining reddening measurements with astrophysical knowledge to determine the host galaxy extinction along the line-of sight: (1) assume that linear $A_V$ is the natural space for extinction and assume a flat prior [@perlmutter99; @knop03]; (2) use models of the dust distribution in galaxies [@hatano98; @commins04; @riello05] to model line-of-sight extinction values [@riess98; @tonry03; @riess04b]; (3) assume that the distribution of host-galaxy $A_V$ follows an exponential form [@jha06c], based on observed distributions of $A_V$ in nearby [SNe Ia]{}; and (4) self-calibrate within a set of low-$z$ [SNe Ia]{} to obtain a consistent color+$A_V$ relationship and assume that relation for the full set [@astier06].

Approach (1) assumes the least prior knowledge about the distribution of $A_V$ and produces a Gaussian probability distribution for the fitted luminosity distance. However, this approach weakens the ability to separate intrinsic [SN Ia]{}color from $A_V$ and results in a fit parameter $A_V$ that is a mixture of the two. An $A_V$ that is truly related to the dust extinction should never be negative. The probability prior with $-\infty < A_V < +\infty$ is not the natural range over which to assume a flat distribution. The physically reasonable prior on $A_V$ should be strictly positive. One approach is to base the prior for absorption on the distribution of dust in galaxies. Theoretical modeling of dust distributions in galaxies, such as that of @hatano98, @commins04, and @riello05, provides a physically motivated dust distribution. This method represents approach (2) above and is the method we adopt here. In contrast, @jha06c empirically derived an exponential $A_V$ distribution from MLCS2k2 fits to nearby [SNe Ia]{} by assuming a particular color distribution of [SNe Ia]{}. This distribution was derived using the empirical fact that [SNe Ia]{}  reach a common color about 40 days past maximum light [@lira95]. They found an exponential distribution of $A_V$, $$p(A_V) \propto \exp \left(\frac{-A_V}{\tau}\right),
\label{eq:default}$$ where $\tau=0.46$ mag. Unfortunately, the highest-extinction objects drive the tail of this exponential and significantly affect the fit, resulting in a prior sensitive to sample selection, which differs significantly in high-redshift searches compared to the nearby objects studied by @jha06c.

A06 analyzed the results of the SALT [SN Ia]{} light-curve fitter with approach (4) and have systematic sensitivities that are similar to those of approach (1).

We use MLCS2k2 as our main analysis tool. We designate approach (1) the “flatnegav” prior and approach (3) the “default” prior and discuss both of these further in §\[sec:priors\]. Approach (2) is based on a galactic line-of-sight or “glos” prior on $A_V$: $$\hat{p}(A_V) \propto \frac{A}{\tau} \exp\left({\frac{-A_V}{\tau}}\right) + 
\frac{2B}{\sqrt{2\pi}\sigma} \exp\left(\frac{-A_V^2}{2\sigma^2}\right),
\label{eq:glos}$$ where $A=1$, $B=0.5$, $\tau=0.4$, $\sigma=0.1$, and $\hat{p}(A_V) \equiv 0$ for $A_V < 0$. This exponential plus one-sided narrow Gaussian “glos” prior is based on the host-galaxy dust models of @hatano98, @commins04, and @riello05. As described below, we have modeled our selection effects with redshift to adapt the “glos” prior into the “glosz” prior that is the basis for our analysis. We feel this approach leverages our best understanding of the effects of extinction and selection.

Figs. \[fig:mlcs\_fits\_prior\_glosz\] and \[fig:salt\_fits\] show the distribution of the fit parameters and overlay the prior distribution assumed for each of these approaches. Fig. \[fig:mlcs\_vs\_salt\] compares the fit distances and extinction/color parameters of the MLCS2k2 “glosz” and SALT fit results for the ESSENCE, SNLS, and nearby samples. The distribution of recovered $\Delta$ and $A_V$ match their imposed priors for MLCS2k2 “glosz” while the stretch and color fit parameters from SALT show a consistent distribution for the three different sets of [SNe Ia]{}.

ESSENCE Selection Effects and the Motivation for a Redshift-Dependent Extinction Prior {#sec:selection}
--------------------------------------------------------------------------------------

We examined the effect of the survey selection function on the expected demographics of the ESSENCE [SNe Ia]{} and explored the interplay between extinction, Malmquist bias, and our observed light curves. To determine the impact of the selection bias, we developed a Monte Carlo simulation of the ESSENCE search. We created a range of supernova light curves that match the properties of the nearby sample, added noise based on statistics from actual ESSENCE photometry, and then fit the resulting light curves in the same way the real events are analyzed. In this way we estimated the impact of subtle biases, although this simulation cannot test for errors in our light-curve model or population drift with redshift.

Based on its low-redshift training set, MLCS2K2 is able to output a finely sampled light curve given a redshift ($z$), distance modulus ($\mu$), light-curve shape parameter ($\Delta$), host extinction ($A_V$), host extinction law ($R_V$), date of rest-frame $B$-band maximum light ($t_0$), Milky Way reddening ($E(B-V)_\mathrm{MW}$), and the bandpasses of the observations. At a given redshift we calculated a distance modulus, $\mu_\mathrm{true}$, from the luminosity distance for the standard cosmology ($\Omega_m=0.3$, $\Omega_\Lambda =0.7$) and that distance modulus plus an assumed $M_B=-19.5$ for [SNe Ia]{} set the brightness for our simulated supernovae. Varying the assumed cosmology does not significantly impact the simulation results since we are comparing the input distance modulus with the recovered distance modulus, $\mu_\mathrm{obs}$, which is independent of the cosmology.

At each of a series of fixed redshifts, we created $\sim1000$ simulated light curves with parameters chosen from random distributions. The light-curve width, $\Delta$, was selected from the @jha06c distribution measured from the low-$z$ sample. The $\Delta$ distribution is approximately a Gaussian peaking at $\Delta =-0.15$ with an extended tail out to $\Delta =1.5$. The host extinction for each simulated event, $A_V$, was selected from either the @jha06c distribution (“default”) estimated from the local sample or from a “galaxy line-of-sight” estimation (“glos”). The “default” distribution was an exponential decay with index $0.46$ mag and set to zero for $A_V < 0.0$ mag. The “glos” distribution is also set to zero for $A_V < 0.0$ mag and combines a narrow Gaussian with a exponential tail for $A_V > 0.0$ mag (see Eq. \[eq:glos\]). The extinction law is assumed to be $R_V = 3.1$. The Milky Way reddening \[$E(B-V)_\mathrm{MW}$\] distribution was constructed from the @schlegel98 (hereafter SFD) reddening maps that cover the ESSENCE fields. The $E(B-V)_\mathrm{MW}$ was measured for 10,000 random locations in each ESSENCE field and the reddening was selected from the sum of the histograms (see Figure 2). The dates of observation for a simulated [SN Ia]{} were based on the actual dates of ESSENCE 4-m observations. An ESSENCE field was chosen at random from the list of monitored fields and a date of maximum, $t_0$, selected to fall randomly between the Modified Julian Date (MJD) of the first and last observation of an observing season. The simulated light curve was then interpolated for only those dates that ESSENCE took images. With each ESSENCE field observation, we estimated the magnitude in $R$ and $I$ that provided a 10$\sigma$ photometric detection based on the seeing and sky brightness. The signal-to-noise ratio (SNR) for each simulated light-curve point was then scaled from the 10$\sigma$ detection magnitude, assuming the noise was dominated by the sky background.

For each date of ESSENCE observation, we have a simulated noiseless magnitude and an estimate of the SNR of the observation. To each simulated observation we added an appropriate random value in flux space selected from a normal distribution with a width corresponding to the predicted SNR.

MLCS2k2 was then used to fit the simulated light curves and provide estimates of $\mu$, $\Delta$, $A_V$, and $t_0$, assuming a fixed $R_V = 3.1$. MLCS2k2 required an initial guess of the date of maximum, an estimate achieved by selecting from a normal distribution about the true date with a $1\sigma$ width of 2 days. The SFD Milky Way reddening was also required in MLCS2k2 and was provided from the true reddening after adding an uncertainty of 10%. Finally, in the real ESSENCE data we discarded supernovae when the MLCS2k2 reduced $\chi^2$ indicated a very poor fit. For the simulated light curves, we dropped events from the sample if the reduced $\chi^2$ exceeded 2.

### Deriving an Extinction Prior from the Simulation Results

Simulated ESSENCE samples were created at a range of redshifts out to $z=0.70$ and the light curves that passed the detection criteria from the actual ESSENCE search were fit with MLCS2k2. The fitting was done with the “default” prior and the “glos” prior (with corresponding $A_V$ distributions). The difference between the “true” (input) distance modulus and recovered (fit) distance modulus, $\Delta\mu$, was calculated for each event and the mean, median, and dispersion for the ensemble were calculated at each redshift. The median $\Delta\mu$ of the simulations was within $0.03$ mag for $z < 0.45$, but at higher redshift the simulated supernovae were estimated to be brighter than the input supernovae by more than $0.2$ mag. This bias results from the loss of faint events (large $A_V$ and large $\Delta$) from the sample as the distance increases. In a sense, this is a classic Malmquist bias, but here it is caused by an uninformed prior. These results are shown in Fig. \[fig:simulatepriors\].

The decreasing ability to observe large $A_V$ events as the redshift increases (see Fig. \[fig:cut\]) makes it clear that a using single $A_V$ prior for all redshifts is not correct. Because events with large $A_V$ and large $\Delta$ are lost at large redshift due to the magnitude limits of the search, we should adjust the prior as a function of $z$ to account for these predictable losses. Applying redshift-dependent window functions to the basic “glos” prior provides a much better prior as a function of redshift.

We fit the recovered $A_V$ distributions derived from the simulations, which start with a uniform $A_V$, to a window function based on the error function (integral of a Gaussian), and two parameters describe where that function drops to half its peak value ($A_{1/2}$) and the width of the transition ($\sigma_A$). The window function $W$ has the form $$W(A_V,A_{1/2},\sigma_A) = 1-
{1\over{\sqrt{\pi}}}\int_{-\infty}^{(A_V-A_{1/2})\sigma_A}
e^{-x^2}\;dx
\label{eq:gloszwindow}$$ where $A_{1/2}$ and $\sigma_A$ are functions of $z$ and estimated from the simulations. A similar process was applied to the $\Delta$ distribution and a table providing the parameters is given in Table \[tab:gloszwindow\]. We embody this prescription in the “glosz” prior we use for our main MLCS2k2 light-curve fitting. The “glosz” prior is the “glos” prior modified by the window functions in $A_V$ and $\Delta$. The simulations using the “glosz” prior provide a median $\Delta\mu$ within 0.03 mag for $z < 0.7$, which we judge to be satisfactory performance.

Comparison of MLCS2k2 and SALT Luminosity Distance Fitters
----------------------------------------------------------

The release of the source code to the SALT fitter [@guy05] makes a modern [SN Ia]{} light-curve fitter fully accessible and available to the community. This public release of SALT allows us to compare the results of our MLCS2k2 distance fitter with the SALT fitter used in the SNLS first results paper [@astier06]. We present the results of SALT fits to our nearby and ESSENCE samples in Table \[tab:salt\_fits\]. To compute the distance moduli we quote in that table, we assume the $\alpha=1.52$, $\beta=1.57$ values from A06. To calibrate the additional dispersion to add to the distance moduli of MLCS2k2 and SALT, we fit a [$\Lambda{\rm CDM}$]{} model to the nearby sample alone and derived the additional [$\sigma_{\rm add}$]{} to added in quadrature to recover [$\chi^2/{\rm DoF}$]{}$=1$ for the nearby sample. This [$\sigma_{\rm add}$]{} is related to the intrinsic dispersion of the absolute luminosity of [SNe Ia]{}, but is not precisely the same both because the light-curve fitters include varying degrees of model uncertainty and because the light curves of the [SNe Ia]{} are subject to photometric uncertainty. We find [$\sigma_{\rm add}$]{}$=0.10$ for MLCS2k2 with the “glosz” prior and [$\sigma_{\rm add}$]{}$=0.13$ for SALT. These values should be added to the $\sigma_\mu$ uncertainties given Tables \[tab:mlcs\_fits\_prior\_glosz\] and \[tab:salt\_fits\] Fig. \[fig:mlcs\_vs\_salt\] visually demonstrates that the relative luminosity distances using the SALT light-curve fitter agree, within uncertainties, with the MLCS2k2 distances when the latter are fit using the “glosz” $A_V$ prior.

Testing the Recovery of Cosmological Models Using Simulations of the ESSENCE Dataset {#sec:simulation}
------------------------------------------------------------------------------------

In order to assess the reliability with which we recover cosmological parameters, we have simulated 100 sets of 100 light curves representing both the nearby and the ESSENCE light curves. Table \[tab:mlcs\_cuts\] presents the quality cuts for MLCS2k2 we derived from these simulated light-curve sets. Our light-curve goodness-of-fit cuts, when applied to these simulated light curves (see Table \[tab:mlcs\_cuts\]) and combined with the same external constraints of baryon acoustic oscillations [BAO; @eisenstein05] and flatness, allow us to recover our input cosmology of $(\Omega_M=0.3, \Omega_\Lambda=0.7, w=-1)$ to within $\pm0.11$ in $w$. This $\pm0.11$ uncertainty on an individual measurement of $w$ is matched by the $\sigma=0.11$ distribution of recovered $w$ values from the 100 sets of simulated light curves. This confirms our statistical error estimate on $w$; the estimated uncertainty matches the distribution, and within the self-consistent realm of synthetic and analyzed light curves based on MLCS2k2 our estimates of luminosity distance are not biased.

Potential Sources of Systematic Error {#sec:systematics}
=====================================

Here we identify and assess sources of systematic error that could afflict our measurements. These can be divided into two groups. Certain sources of systematic error may introduce perturbations either to individual photometric data points or to the distances or redshifts estimated to the [SNe Ia]{}. Others affect the data in a more or less random fashion and produce excess [*scatter*]{} in the Hubble diagram. Errors that are uncorrelated with either distance or redshift will not bias the cosmological result. These sources of photometric error are detailed by @miknaitis07; we summarize those results here in Table \[tab:photscatter\]. We add these effects in quadrature to the statistical uncertainties given by the luminosity distance fitting codes for each [SN Ia]{} distance measurement: $\sigma_\mu^{\rm phot scatter}=0.026$ mag.

In §\[sec:distances\] we discussed our testing of the MLCS2k2 fitter on simulated data sets that replicate the data quality of the ESSENCE and nearby [SNe Ia]{}. We explore the issue of host-galaxy extinction further in §\[sec:extinction\] & \[sec:priors\]. The interaction of Malmquist bias and selection effects with the extinction and color distribution of [SNe Ia]{}is discussed in §\[sec:malmquist\].

Any non-cosmological difference in measurements of nearby and distant [SNe Ia]{} has the potential to perturb our measurement of $w$. Table \[tab:systematics\] lists potential systematic effects of this sort. We present both our estimate of the sensitivity $dw/dx$ of the equation-of-state parameter to each potential systematic effect and our best estimate of the potential size of the perturbation, ${\Delta}x$. The upper bound on the bias introduced in $w$ is then ${\Delta}w = dw/dx\times{\Delta}x$. @miknaitis07 discusses the systematic uncertainties on $\mu$, which we convert here to systematic uncertainties on $w$, due to photometric errors from astrometric uncertainty on faint objects (${\Delta}w = 0.005$), potential biases from the difference imaging (${\Delta}w = 0.001$), and linearity of the MOSAIC II CCD (${\Delta}w = 0.005$). None of these contributed noticeably to the systematic uncertainty in our measurement of $w$. The rest of this section describes how we appraised our additional potential sources of systematic uncertainty.

The conclusion of this section is that our current overall estimate for the 1$\sigma$ equivalent systematic uncertainty in a static equation-of-state parameter is ${\Delta}w =$ [$0.13$]{} for our “glosz” analysis.

Photometric Zeropoints {#sec:zeropoint}
----------------------

Supernova cosmology fundamentally depends on the ability to accurately measure fluxes of objects over a range in redshift. Errors in photometric calibration translate to errors in cosmology in two basic ways.

Nearby objects at redshifts $<0.1$ play a crucial role in establishing a comparison reference for cosmological measurements. ESSENCE is inefficient at finding and observing low-redshift objects with the same telescope and detector system, so we use photometry of low-redshift [SNe Ia]{} in the literature from our own work and that of others [for the full list see @jha06c]. Using these external [SNe Ia]{} requires understanding the photometric calibration of our high-redshift sample relative to this low-redshift sample. Every supernova cosmology result to date has made use of more or less the same low-redshift photometry, so any inaccuracies in the nearby sample are a source of common systematic error for all [SN Ia]{} cosmology experiments. Calibration of photometry at the $\sim1\%$ level required to make precise inferences about the nature of dark energy is notoriously difficult [@stubbs06].

Photometric miscalibration can result in a second, more insidious systematic error if there is an error in the relative flux scaling between the broad-band passbands. This offset would distort the observed colors for the entire sample. Since these colors are used to infer the extinction, even small color errors result in significant biases in the measured distances. After all, the inferred host galaxy extinction, $A_{V}$, is related to the measured color excess, $E(B-V)$, by $A_{V}\approx3.1 E(B-V)$ (for Milky Way-like dust). A color error in rest-frame $B-V$ (observer-frame $R$, $I$ for ESSENCE) of $0.01$ mag can result in $0.03$ mag error in extinction, an inaccuracy that would lead directly to a $3\%$ error in the distance modulus, or a $1.5\%$ error in the distance. We currently estimate our color zeropoint uncertainty at $0.02$ mag and our absolute zeropoint (relative to the nearby [SNe Ia]{}) uncertainty to be $0.02$ mag. These respectively translate to $0.04$ and $0.02$ shifts in $w$ (see Table \[tab:systematics\]).

@miknaitis07 describe the calibration program we undertook to measure the transmission of the CTIO 4-m MOSAIC II system with the $R$ and $I$ filters of the ESSENCE survey. The calibration of the ESSENCE survey fields will be further improved by an intensive calibration program we are undertaking on the CTIO 4-m in 2006. Together with the improved calibration of the SDSS Southern Stripe by the SDSSII project, which overlaps 25% of our ESSENCE fields, we aim to achieve 1% photometric calibration of our CTIO 4-m MOSAIC II BVRI natural system.

We here use MLCS2k2 v004 with the @bohlin04 values for the magnitudes of Vega: i.e., `alpha_lyr_stis_002.fits` with $R_{\rm
Vega}=0.033$ mag. This value for $R_{\rm Vega}$ comes from @bessell98 but has been shifted down by $0.004$ mag as @bohlin04 suggest (from their $V_{\rm Vega}=0.026$ mag compared to @bessell98 $V_{\rm Vega}=0.030$ mag).

K-Corrections and Bandpass Uncertainty {#sec:kcorrection}
--------------------------------------

Uncertainty in the transmission function, typically called the bandpass, of the optical path of the telescope+detector is an important and potentially systematic effect. In this context, bandpass refers to the wavelength-dependent throughput of the entire optical path, including atmospheric transmission, mirror reflectivity, filter function, and CCD response. Since an error in the assumed bandpasses translates into a redshift-dependent error in the supernova flux, it is important to account for possible errors in the bandpass estimates.

The *relative* error due to bandpass miscalibration is small for objects with similar spectra, such as [SNe Ia]{}. Bandpass shape errors are largely accounted for by the filter zeropoint calibration, with residual errors corresponding to the difference between the spectral energy distribution of the objects of interest and those of the calibration sources. In the case of [SN Ia]{} observations, any residual zeropoint error is absorbed when we marginalize over the “nuisance parameter,” [$\cal{M}$]{}$=M_B - 5\log_{10} (H_0) + 25$ [@kim04]. This relative comparison results in a very small systematic error in the cosmological parameters from a global calibration error across bandpasses. Moreover, variations in atmospheric transmission are expected to contribute only random uncertainty.

However, the bandpass uncertainty becomes important when we compare [SNe Ia]{} at different redshifts for which the bandpass samples different spectral regions. In order to compare [SNe Ia]{} at multiple redshifts, we need to perform a K-correction [@leibundgut90; @hamuy93a; @kim96; @nugent02]. That is, we assume a spectral distribution for the supernova and convert the observed magnitude to what it would have been had the supernova been at another redshift. This process involves performing synthetic photometry of the assumed spectral distribution over the assumed bandpass. We address the issue of systematics arising from errors in the assumed spectral distribution in the supernova evolution section, §\[sec:snevolution\]. Here we address systematics arising from errors in our determination of the CTIO 4-m MOSAIC II $R$ and $I$ bandpass functions. Systematic effects on supernova cosmology that result from bandpass uncertainties are discussed more thoroughly by @davis06.

\[sec:bandpass\]

Calculating the effect of bandpass uncertainty is fairly difficult because of the arbitrary nature of the shape changes that might affect the bandpass. However, we can make several general calculations. As a first step, we take standard bandpasses and add white noise to represent a miscalibrated filter. White noise contributes power on all scales, so this approach adds small-scale discrepancies as well as large-scale warps or shifts in the filter. By averaging over many such miscalibrated filters, we can estimate the effect of filter miscalibration. Fig. 16 of @davis06 shows photometric error as a function of noise amplitude. A noise amplitude of $0.02$ produces a typical deviation of $2$% from the nominal filter shape at any wavelength. Calibrating the bandpass to better than $3\%$ allows us to keep the K-correction error introduced from a mismeasurement of our effective bandpass to less than 0.005 mag (0.5% in flux) and a systematic uncertainty of ${\Delta}w = 0.005$.

Extinction {#sec:extinction}
----------

The most significant cause of variation in luminosity of [SNe Ia]{} is the extinction experienced by the light from the [SN Ia]{}due to scattering and absorption from dust in the host galaxy.

Dust introduces a wavelength-depended diminution of a supernova’s light. In the case of Milky Way dust, we correct for its effects by using tabulated values as a function of Galactic longitude and latitude measured by other means [SFD @schlegel98], being sure in our MLCS2k2 fits to properly account for its uncertainty and correlation across all observations. For dust in the supernova’s host galaxy, we infer the extinction from the reddening of each supernova’s light curve.

However, the slope of differential reddening, characterized in the @cardelli89 extinction model by the parameter $R_V$, may vary. The nominal value of $R_V$ for the Milky Way is $3.1$, but different lines of sight within our galaxy have values of $R_V$ that vary from $2.1$ to $5.1$. Studies of $R_V$ in other galaxies have been more limited because we lack sources of known color and luminosity with which to probe the dust.

Because we use the supernova rest-frame $B-V$ color to determine the reddening of each [SN Ia]{}, and the distance modulus to a supernova is corrected by a value approximately three times the inferred reddening, extinction correction magnifies any source of systematic error in a supernova’s observed effective color. Systematic color errors can result from photometry errors, redshift-dependent K-correction errors, and evolution in the colors of supernovae.

Using the IR-emission maps of the Galaxy from the all-sky COBE/DIRBE and IRAS/ISSA maps, SFD have estimated the dust column density around the sky, which can then be translated to a color excess. This analysis has largely superseded the work of @burstein78, who used radio HI measurements and a relationship between gas and extinction to estimate the color excess across the sky. @burstein03 has reanalyzed the IR and HI measurements and finds that Milky Way extinctions are more precisely derived using the IR method. However, @burstein03 still finds a discrepant value for extinction at the poles, with SFD providing extinctions that are $E(B-V)=0.02$ mag higher than what the HI measurements indicate. @burstein03 suggests as a possible explanation for the discrepancy that SFD may predict too large an extinction in areas with high gas-to-dust ratios. @finkbeiner99 precisely estimated their sensitivities to these systematics and concluded they had controlled them to $0.01$ mag. The ESSENCE program targets fields at high Galactic latitude to minimize Galactic extinction. Although nearby and distant [SNe Ia]{}are both affected by the assumed Milky Way extinction, the nearby objects are observed in $B-V$, whereas the $z\approx0.5$ objects are observed in $R-I$. An $E(B-V)=0.02$ difference in extinction at the pole leads to approximately a $0.02$ mag difference in the relative distances between $z=0$ and $z=0.5$ objects, assuming a Galactic reddening law, host-galaxy corrections based on rest-frame $B-V$ color, and distances based on $V$. For this analysis, we use the SFD extinction map values with an uncertainty of 16% for each individual [SN Ia]{}  but assume an additional $0.01$ mag of systematic uncertainty in our distance moduli to account for the known source of uncertainty of extinction at the pole.

In most supernova work we assume the Galactic reddening law [@cardelli89] applies to external galaxies ($R_V=3.1$), but studies of individual [SNe Ia]{} have found a range of values extending to much smaller values of $R_V$ [@riess96; @tripp98; @phillips99; @krisciunas00; @wang03; @altavilla04; @reindl05; @elias-rosa06]. These measurements are dominated by objects with large extinction values, where a significant measurement can be made can be made of the extinction law (lessening the effects of intrinsic color scatter and systematic color variations with luminosity), and it is possible that $R_V$ is correlated with total extinction [@jha06c]. In principle, with photometry in three or more passbands, it is possible to fit for $R_V$, but in practice, at $z>0.2$, there are only a few [SNe Ia]{} in the literature with the requisite high-precision photometry extending from the rest-frame UV to the near-IR. The systematic error on our measurement of $D_L$ caused by assuming a particular value of $R_V$ depends on the average extinction as a function redshift, assuming $R_V$ is constant with $z$, except for a small correction caused by the rest-frame effective bandpass of our filters drifting away from the low-$z$ values, depending on the precise redshift of each object. To quantify this effect, we fit our complete distance set with three different values of $R_V$: $2.1$, $3.1$, and $4.1$.

Color and Extinction Distributions and Priors {#sec:priors}
---------------------------------------------

To evaluate the systematic effects produced by various prior assumptions about extinction, we have fit the entire data set with a variety of plausible priors: the “exponential” prior of @jha06c, a flat prior from $-\infty$ to $+\infty$ (the “flatnegav” prior), and an exponential prior with an added Gaussian around zero that is based on models of the dust distribution in galaxies (“glos” and the redshift-dependent “glosz”). These results are presented in §\[sec:cosmology\] and form the basis for Table \[tab:w\_rv\_sys\].

To separate the effects of color and extinction, @jha06c noted that the distribution of color excess in their nearby sample was consistent with a Gaussian distribution of $\sigma=0.2$ convolved with a one-sided exponential, $\exp{(-A_V/\tau)}$, where $\tau=0.46$ mag. As discussed in §\[sec:extinction\_prior\], the “glosz” prior we adopt here is derived from models of line-of-sight dust distributions in galaxies. It has more parameters than the simple exponential model of @jha06c, but we believe these additional parameters are well motivated.

The power of MLCS2k2 to distinguish between color and extinction lies in the ability to treat the two phenomena independently. A06 uses SALT and makes the assumption that the color$+$extinction distribution is the same in the nearby and in the high-redshift samples; the separation of the $A_V$ component in the MLCS2k2 model allows us to model our expected distribution of $A_V$ based on both models of dust in galaxies and selection effects of the ESSENCE survey. This separation allows us to take the nominal “glos” model and create the “glosz” prior that combines the distribution of dust in galaxies with the redshift-dependent selection effects.

The difference in the mean estimated parameter for a constant $w$ is given in Table \[tab:w\_rv\_sys\] for the different MLCS2k2 $A_V$ priors discussed above. For the main MLCS2k2 “glosz” analysis we present here, we find a slope of ${\Delta}w / {\Delta}R_V=0.02$ in the dependence of $w$ on the assumed $R_V$. The effect on $w$ of varying $R_V$ is substantially greater for the less restrictive $A_V$ priors because the covariance between $A_V$ and $\mu$ is substantially greater for these priors. A reasonable variation of $0.5$ in the value for $R_V$ contributes a systematic uncertainty of ${\Delta}w=0.01$

Differences in the inferred value of $w$ for various assumed absorption priors shows that this is a significant systematic effect. The maximum difference between two priors, “exponential” and “glosz,” for the nominal $R_V=3.1$ case is ${\Delta}w=0.165$. While we have conducted careful simulations to determine the most appropriate prior for our sample (see §\[sec:selection\]) and it is clear that the “exponential” is not appropriate for this analysis, we nonetheless take half of the difference between the two as representative of our systematic uncertainty, $\Delta_w^{\rm prior}=0.08$, due to the choice of prior. The residual $0.02$ mag shift of the simulations with the “glosz” prior shown in Fig. \[fig:cut\] for $z\approx0.65$ results in a very small shift in ${\Delta}w$ of only $0.001$. Since we use an $A_V$, that obviously interacts strongly with our understanding of the intrinsic color distribution of [SNe Ia]{}. We estimate this contribution to our systematic error budget at ${\Delta}w=0.06$ We have not undertaken a similar analysis with the SALT fitter, but the underlying assumption that the color, extinction, luminosity relationship for [SNe Ia]{} is constant with redshift is subject to uncertainties analogous to those considered here in the context of the MLCS2k2 $A_V$ prior. The issue of color and extinction distributions clearly needs to be addressed for substantial further progress to be made in the field of supernova cosmology.

Malmquist Bias and Other Selection Effects {#sec:malmquist}
------------------------------------------

As with all magnitude-limited surveys, at the faint limits of the survey we are more likely to observe objects drawn from the bright end of the [SN Ia]{} luminosity distribution. This Malmquist bias is particularly dangerous for inferences about cosmology based on supernova observations. However, it is not necessarily troubling that we may observe more luminous, broad events at high redshift, as long as the known empirical luminosity-width relation is valid at those redshifts. Rather, the concern for cosmological measurements is that at high redshift, we may preferentially find [SNe Ia]{} which are bright *for their light curve shape*. A second and more subtle concern is that at higher redshifts we are also less likely to detect [SNe Ia]{} whose light suffers significant absorption due to dust in their host galaxies.

We have modeled both of these effects (see §\[sec:selection\] & \[sec:extinction\]) and have controlled for their impact. Our current limits on systematics due to uncontrolled selection effects is $\Delta_w^{\rm selection}=0.02$. A thorough study of the efficiency of the ESSENCE survey will be presented by @pignata07. We aim for this future work to allow us to reduce this contribution to our systematic error to no more than 1%.

Type Ia Supernova Evolution {#sec:snevolution}
---------------------------

A persistent concern for any standard-candle cosmology is the possibility that the distant candles may differ slightly from their low-redshift counterparts. In a recent paper [@blondin06] we compare the spectra of the high-redshift [SNe Ia]{} in this sample with low-redshift [SNe Ia]{} and demonstrate that there is no evidence for any systematic difference in their properties. This conclusion is based on line-profile morphology and measurements of the phase-evolution of the velocity location of maximum absorption and peak emission.

These results confirm a number of other studies of distant [SNe Ia]{}[e.g., @coil00; @sullivan03; @lidman04] that all confirm that, to the accuracy of current observations, the high and low redshift supernova populations are indistinguishable. Recently @hook05 used spectral dating, spectral time sequences, and measurements of expansion velocities to compare distant and nearby [SNe Ia]{}; they also find no evidence for evolution in [SN Ia]{} properties up to $z\approx0.8$.

Although we are confident that the subtypes of distant [SNe Ia]{} are well represented by the subtypes seen nearby, we cannot rule out a subtle shift in the population demographics that may yet bias the estimates of cosmological parameters. This potential bias is of particular concern for future experiments that plan to measure the equation-of-state parameter, $w$, with an accuracy of a few percent. There is now evidence that [SN Ia]{} properties are correlated with host-galaxy morphology. @hamuy96 and @riess99 show that the brightest [SNe Ia]{} occur only in galaxies with on-going star formation. However, they observe no residual correlation after light-curve shape correction. Because the galactic demographics over the redshift range of interest change less than current variations in stellar population of [SN Ia]{}host galaxies, we remain confident that our one-parameter correction for supernova luminosity adequately corrects any shift in the average luminosity of [SNe Ia]{} to the same precision as in the nearby Universe, $\sigma_\mu < 0.02$ mag. We thus estimate a systematic uncertainty from possible [SN Ia]{}evolution on our measurement of $w$ of ${\Delta}w=0.02$.

One way to verify this confidence is to search for additional parameters that allow tighter luminosity groupings of the low-redshift population. In a first, reassuring step, Hubble diagrams for subsets of [SNe Ia]{} based on host-galaxy type separately confirm the accelerated expansion of the Universe [@sullivan03].

Hubble Bubble and Local Large-Scale Structure {#sec:bubble}
---------------------------------------------

The local large-scale structure and associated correlated flows of the Universe should not yet present a significant contribution to the systematic error budget of the current survey [@hui06; @cooray06]. However, at the lowest multipoles we are sensitive to local correlated flows, and, at the most extreme, our cosmological results would be sensitive to a local velocity monopole or “Hubble bubble.” @jha06c see such an effect in their analysis of nearby [SNe Ia]{}. We use only the subset of [SNe Ia]{} from @jha06c with $z>0.015$ and find that this effect could contribute as much as $0.065$ to our systematic error budget in $w$. We will rely on future sets of nearby [SNe Ia]{} ($0.01<z<0.05$) that are now being acquired at the CfA, by the Carnegie Supernova program, by the Lick Observatory Supernova Search, and by the SNfactory to reduce this uncertainty below 2% to help achieve the desired systematic uncertainty required for the final ESSENCE analysis.

Gravitational Lensing {#sec:lensing}
---------------------

Gravitational lensing can increase or decrease the observed flux from a distant object. The expected distribution is asymmetric about the average flux multiplier of unity. @holz05 calculate the effect for [SN Ia]{}surveys and determine that any systematic effect from neglecting the asymmetry of the probability distribution function for magnification (as we do here) decreases quickly with the number of [SNe Ia]{} per effective bin. Roughly speaking, at a $z\approx0.5$, in a redshift bin width of ${\Delta}z \sim 0.1$, ten [SNe Ia]{} per bin are sufficient to reduce any systematic effect in luminosity distance to less than $0.3\%$, which makes no noticeable contribution to our systematic error budget. For the redshifts of interest in the ESSENCE survey, lensing has a more significant effect in the scatter it adds to the observed brightness of [SNe Ia]{}. @holz05 calculate a 3% increase in the dispersion in distance modulus at $z\approx0.5$. We include the effect of lensing in our analysis by adding a statistical dispersion of $\sigma^{\rm lensing}_{\mu}=0.03$ to our luminosity distance uncertainty for the ESSENCE and SNLS [SNe Ia]{}.

Grey Dust {#sec:greydust}
---------

When the first cosmological results with [SNe Ia]{} were announced, that distant [SNe Ia]{} were dimmer than they would be in a decelerating Universe, @aguirre99a [@aguirre99b] suggested various models for intergalactic grey dust that could explain this dimming without producing observable reddening. To explain [SNe Ia]{}becoming consistently dimmer with distance, this dust would need to be distributed throughout intergalactic space beginning at least at $z=2$ [@goobar02]. The most naive model of such dust distribution and creation would predict that [SNe Ia]{} should continue to get dimmer relative to a flat, [${\Omega}_{\rm M}$]{}$=1$, cosmology all the way up to at least a redshift of $2$. The high-redshift [SN Ia]{} work of @riess04b demonstrated that this continued dimming is not what is observed: the apparent magnitudes of [SNe Ia]{} become first a little dimmer and then a little brighter with redshift than they would in an empty Universe. This is exactly what we expect from an early phase of deceleration followed by a recent phase of acceleration in a mixed, dark-matter/dark-energy cosmology.

A more complicated model of dust was contrived by @goobar02. It involves the creation of intergalactic dust at just the right rate to match the decrease in opacity due to expansion of the Universe. This carefully constructed model mimics the signal of an accelerating universe and is difficult to distinguish from a universe that is presently dominated by dark energy. This model does not have a strong underpinning in the behavior of known dust and represents a form of fine-tuning. In the larger context of converging cosmological evidence, this particular scheme for matching the data seems less plausible than a universe with dark energy.

Recent observational constraints from non-[SN Ia]{} sources have independently placed significant constraints on the amount of intergalactic dust [@petric06; @ostman06]. In particular, the observations of @petric06 limit intergalactic dust to contributing no more than one percent to potential dimming of light out to a redshift of $0.5$, based on upper limits to X-ray scattering by dust along the line of sight to a quasar at $z=4.3$.

Cosmological Results from the ESSENCE Four-Year Data {#sec:cosmology}
====================================================

The ESSENCE [SNe Ia]{} allow us test the hypothesis of a [$\Lambda{\rm CDM}$]{} concordance model and constrain flat, constant-$w$ models of the Universe. We use our MLCS2k2 light-curve fitting technique to measure luminosity distances to nearby and ESSENCE [SNe Ia]{} (Table \[tab:mlcs\_fits\_prior\_glosz\]). When then fit cosmological models to constrain the properties of the dark energy. We compare the results we obtain using MLCS2k2 with those obtained using the SALT light-curve fitter [@guy05]. The SALT fitter was used to fit the nearby light curves, our ESSENCE light curves, and the SNLS light curves.[^4] To verify that we were making appropriate use of the fitter, we fit the nearby and SNLS light curves with SALT, taking the same $\alpha=1.52$ and $\beta=1.57$ width and color parameters used in A06. We recovered the $w$ result of A06 to within $0.01$ in best-fit constant $w$ in a model with a flat Universe using the cosmology fitter that we employ here[^5]. We have compiled our light curves of nearby [SNe Ia]{} from the literature independently of the SNLS analysis and used slightly different quality cuts, so it is quite encouraging that we can replicate these results. Table \[tab:salt\_fits\] gives the SALT fit parameters for the nearby, ESSENCE, and SNLS [SNe Ia]{} discussed here.

ESSENCE [SN Ia]{} Sample {#sec:essence}
------------------------

For the ESSENCE project we find that using photometric selection criteria based on the color and rise time of the candidate object, similar to those used by the SNLS [@howell05; @sullivan06a], and in good weather and seeing conditions, 80% of the candidates we take spectra of are [SNe Ia]{}. We use a deterministic analysis [@blondin07], as described in @miknaitis07, to determine final types and redshifts for our SNe and to cull objects that are not [SNe Ia]{} from our sample. All of the ESSENCE supernovae used in this analysis were spectroscopically confirmed as [SNe Ia]{}.

From 2002–2005 the ESSENCE project discovered and spectroscopically confirmed 113 [SNe Ia]{}. As discussed by @miknaitis07, which gives full details of these [SNe Ia]{} including their RA and Dec, only 4 of the 15 [SNe Ia]{} from 2002 have been fully analyzed so that leaves us with 102 [SNe Ia]{}. Although we kept 91T-like [SNe Ia]{} such as d083, d085, and d093, we rejected the peculiar [SN Ia]{} d100 [@matheson05]. Three [SNe Ia]{} were rejected from the nearby+ESSENCE only fits because they were at redshifts greater than $0.67$ (see below). After we applied the cuts in Tables \[tab:mlcs\_cuts\] and Tables \[tab:salt\_cuts\], we were left with 57 and 60 [SNe Ia]{} for MLCS2k2 and SALT respectively.

With the MLCS2k2 fitter, the largest cut was the 32 [SNe Ia]{} rejected because they had fewer than 8 data points with an SNR $> 5$, no such points after +9 days, or no such points before +4 days. Two of the 102 [SNe Ia]{} were located near edges of the detector field-of-view that we later determined were photometrically less reliable. Due to high [$\chi^2/{\rm DoF}$]{} or related poor light-curve goodness-of-fit values, we eliminated an additional 6 [SNe Ia]{}. This left us with a total of 57 [SNe Ia]{} for our main MLCS2k2 nearby+ESSENCE analysis. The SALT fitter successfully fit three more [SNe Ia]{} than MLCS2k2, but, in general, our SALT quality cuts accepted the same [SNe Ia]{} as our MLCS2k2 quality cuts. The requirements we imposed here on the light curves were stringent cuts to ensure reliable fit parameters. We are currently engaged in an active program to improve the sensitivity of [SN Ia]{} light-curve fitters and we anticipate recovering 50% of the [SNe Ia]{} rejected here in the final ESSENCE analysis.

Nearby [SN Ia]{} Sample {#sec:nearby}
-----------------------

The [SN Ia]{} cosmological measurement is fundamentally a comparison of the luminosity distance vs. redshift relation at low redshift and high redshift. The ESSENCE [SNe Ia]{} alone provide a homogeneous set of luminosity distance vs. redshift measurements covering the redshift range $0.15<z<0.7$. We selected our nearby [SNe Ia]{} from the set compiled by @jha06c. We applied the light-curve criteria from Tables \[tab:mlcs\_cuts\] and \[tab:salt\_cuts\] and also rejected known peculiar [SNe Ia]{} such as SN 2000cx [@li01] and SN 2002cx [@li03; @jha06b]. Our list of nearby [SNe Ia]{} has 41 [SNe Ia]{} in common with the set used by A06. To minimize complications from loosely constrained local peculiar and coordinated flows, we only considered [SNe Ia]{} with CMB-frame redshifts of $z>0.015$. Our final sample consists of [45]{} nearby [SNe Ia]{} as listed in the fit parameter tables (Tables \[tab:mlcs\_fits\_prior\_glosz\] and \[tab:salt\_fits\]). We used the re-derived Landolt/Vega calibration of A06 to determine the passbands for this set of nearby [SNe Ia]{}. The light curves we used for these [SNe Ia]{} are also included with the ESSENCE light curves available on our website.[^6]

External Constraints
--------------------

To provide complementary cosmological constraints on our [SN Ia]{} observations, we include external information from baryon acoustic oscillations [BAO; @eisenstein05]. The BAO constraints on ([${\Omega}_{\rm M}$]{}, $w$) from @eisenstein05 are the most complementary measurement in the ([${\Omega}_{\rm M}$]{}, $w$) plane to our [SN Ia]{} measurements, relying only on the observed redshift and angular size of the first doppler peak in the CMB and not on $H_0$. In addition, because the BAO constraints on [${\Omega}_{\rm M}$]{} are similar in precision (and value) to those derived from large scale structure [@percival01; @percival02], WMAP directly [@spergel06], and from the study of X-ray clusters [for a review see @voit05], we choose to combine our results only with the BAO results.

We compare the specific model of a flat Universe with either $w=-1$ or constant $w$ of any value to our data. [SNe Ia]{} have very little leverage on the global flatness of the Universe because they effectively measure the difference between [${\Omega}_{\rm M}$]{} and [${\Omega}_{\Lambda}$]{}, and flatness depends on the sum. @eisenstein05 have constrained curvature to be within [${\Omega}_{\rm K}$]{}$=\pm0.01$ of flat. The results presented here (from the [SNe Ia]{}) on $w$ are not significantly affected by variation of [${\Omega}_{\rm K}$]{} by this amount, because the effects of curvature are not noticeable until looking back to much higher redshift. However, non-flat models will significant alter the BAO results on ([${\Omega}_{\rm M}$]{}, $w$) and therefore our joint constraints.

For our analysis of the ESSENCE and nearby [SNe Ia]{}, we have chosen to additionally limit our redshift range to $z<0.670$ to avoid using the rest-frame $U$ band. Since this remove just three ESSENCE [SNe Ia]{} from our sample, the tradeoff is worthwhile to minimize this source of uncertainty (see §\[sec:distances\]). When we add in the SNLS or Riess gold samples, we relax this constraint to incorporate those higher-redshift [SNe Ia]{}.

In Figs. \[fig:joint\_mlcs\_prior\_glosz\_hubble\_diagram\] and \[fig:joint\_salt\_hubble\_diagram\] we show Hubble diagrams of the nearby, ESSENCE, and SNLS samples for the two different fitters we consider in this paper. We overplot an empty Universe ([${\Omega}_{\rm M}$]{},[${\Omega}_{\Lambda}$]{},$w$) = $(0,0,-1)$, a matter-only open Universe $(0.3,0,-1)$, and a [$\Lambda{\rm CDM}$]{} concordance cosmology $(0.27,0.73,-1)$. The residuals in luminosity distance are then shown with respect to the [$\Lambda{\rm CDM}$]{} model. MLCS2k2 appears to be more suited for the ESSENCE data sample than SALT, although the latter benefits from its flux-based fitting by being able to extract useful luminosity distances from a few more [SNe Ia]{}. One [SN Ia]{}, “d083,” is a particular outlier in both fitters at $\sim0.5$ mag brighter than expected in the best-fit or [$\Lambda{\rm CDM}$]{} cosmologies. @matheson05 found the spectrum of this object to be like that of SN 1991T, which is the archetype of over-luminous [SNe Ia]{}. This [SN Ia]{} is likely an interesting object worthy of further study and is potentially similar to a similarly super-luminous object, SN 2003fg, found in the SNLS survey [@howell06]. However, given that our sample comprises [60]{} objects, we certainly allow for the reasonable statistical possibility of a 3$\sigma$ outlier such as “d083” and thus retain it in our sample.

In Fig. \[fig:mlcs\_prior\_glosz\_salt\_om\_w\] we show the 1$\sigma$, 2$\sigma$, and 3$\sigma$ probability contours for our measurement of $w$ vs. [${\Omega}_{\rm M}$]{} for ESSENCE+nearby alone, the BAO constraints from @eisenstein05, and the combination of the [SN Ia]{} and BAO constraints.

Table \[tab:results\] shows the cosmological parameters $w$ and [${\Omega}_{\rm M}$]{} for each of these sets for flat models of the Universe with a constant $w$ as well as the [$\chi^2/{\rm DoF}$]{} for a concordance cosmology and the 1-D marginalized values. A [$\Lambda{\rm CDM}$]{} model of the Universe fits the MLCS2k2-analyzed ESSENCE+nearby sample with a [$\chi^2/{\rm DoF}$]{} of [$0.96$]{}and a residual standard deviation of [$0.20$]{} mag. Thus, while the estimated $w$ parameter in the constant-$w$ models is $w=$[$-1.05^{+0.13}_{-0.12}~{\rm (stat}~1\sigma{)} \pm 0.13~{\rm (sys)}$]{}, a flat, $w=-1$ model of the Universe is consistent with our data.

Our results from these [60]{} [SNe Ia]{} from the ESSENCE survey are consistent with the results of A06. It is reassuring that two independent teams using different telescopes and studying different regions of the sky find that [SNe Ia]{} at high redshift exhibit the same luminosity distance vs. redshift relationship. These samples strengthen and extend the evidence from [SNe Ia]{} for dark energy and, together with complementary constraints on [${\Omega}_{\rm M}$]{}, point toward simple [$\Lambda{\rm CDM}$]{} models for our Universe.

Joint ESSENCE+SNLS Cosmological Constraints
-------------------------------------------

A new opportunity presents itself with the release of the SNLS light curves from A06 and the light curves presented in this paper. For the first time it is possible to do a proper, self-consistent joint fit of two large, independent sets of distant [SNe Ia]{}.

When fitting the SNLS [SNe Ia]{} with MLCS2k2 and the “glosz” prior we shift the assumed $A_V$ and $\Delta$ prior selection window functions by ${\Delta}z=+0.20$ to represent the greater depth of the SNLS survey. The proper way to derive this prior for SNLS would be to model the SNLS survey efficiency and and fit simulated [SNe Ia]{} with MLCS2k2 as we presented in §\[sec:selection\] for the ESSENCE survey. Similar concerns apply for possible selection effects in the heterogeneously nearby sample. Nevertheless, we believe our use of the “glosz” prior is appropriate for the low-redshift sample (where it is just the “glos” prior) and the simple extension in redshift to be a reasonable approach for the SNLS sample. The additional systematic errors introduced by this joint comparison center on the photometric calibration of the distant sample relative to the nearby [SNe Ia]{}. We estimate that uncertainty to be the same as the calibration uncertainty to the nominal Vega system used by each project: $\Delta{\rm zpt}=0.02$ mag. We have not modeled different offsets between the two data sets, but merely express the uncertainty as an additional uncertainty in our inferred cosmological parameters. This relative zeropoint uncertainty adds an additional ${\Delta}w=0.02$ to our overall systematic uncertainty on $w$.

With our combined analysis, we start with the traditional [${\Omega}_{\rm M}$]{}-[${\Omega}_{\Lambda}$]{}contour plot that was the first clear evidence for dark energy.

Table \[tab:joint\_results\] shows the cosmological parameters $w_0$ and [${\Omega}_{\rm M}$]{} for each of these sets for flat models of the Universe with a constant $w$ as well as the [$\chi^2/{\rm DoF}$]{} for a concordance cosmology. A [$\Lambda{\rm CDM}$]{} model of the Universe fits the SNLS+ESSENCE+nearby sample analyzed using MLCS2k2 “glosz” with a [$\chi^2/{\rm DoF}$]{} of [$0.90$]{} from [162]{} [SNe Ia]{}and a residual standard deviation of [$0.23$]{} mag. A joint analysis of the luminosity distances from the SALT fitter results in a [$\chi^2/{\rm DoF}$]{} of [$2.76$]{} from [178]{} [SNe Ia]{}and a residual standard deviation of [$0.28$]{} mag.. Fig. \[fig:joint\_mlcs\_prior\_glosz\_salt\_om\_w\] show the joint MLCS2k2 and SALT results for this joint sample. The estimated $w$ parameter in the constant-$w$ models is $w=$[$-1.07^{+0.09}_{-0.09}~{\rm (stat}~1\sigma{)} \pm 0.13~{\rm (sys)}$]{}, and a flat, $w=-1$ model of the Universe remains consistent with the current generation of [SN Ia]{} data.

Joint ESSENCE+SNLS+Riess Gold Sample Cosmological Constraints
-------------------------------------------------------------

In order to explore models with varying $w$, we now include the gold sample from @riess04b to extend our reach out to $z\approx1.5$. The high-quality intermediate-redshift samples of the ESSENCE and SNLS surveys provide an excellent complement to the high-redshift [SNe Ia]{}in this set. The heterogeneous nature of the collection of [SNe Ia]{} in the gold sample makes it beyond the scope of this paper to produce definite estimates of the systematic errors that result from including this additional set, but it is tempting to add these [SNe Ia]{} and examine the new constraints on cosmological parameters.

We used the 39 nearby [SNe Ia]{} in common between the nearby [SN Ia]{} sample we discuss here and the gold sample to normalize the luminosity distances between the two sets. To avoid double-counting of [SNe Ia]{} in this joint analysis, we then drop the nearby [SNe Ia]{} from the gold sample and use only the nearby [SNe Ia]{} fit in this paper.

We first compute the [${\Omega}_{\rm M}$]{}-[${\Omega}_{\Lambda}$]{} contours to update the case for dark energy from [SNe Ia]{}. Fig. \[fig:OMOL\_OMw\_joint\_riess04\] represents the most stringent demonstration to date of the existence of dark energy. The [SNe Ia]{} data alone rule out an empty Universe at $4.5$ $\sigma$, an ([${\Omega}_{\rm M}$]{}, [${\Omega}_{\Lambda}$]{}) = $(0.3, 0)$ Universe at $10$ $\sigma$, and an ([${\Omega}_{\rm M}$]{}, [${\Omega}_{\Lambda}$]{}) = $(1, 0)$ $\sigma$ Universe at $>20$ $\sigma$. The joint constraints on constant-$w$ models from this full set are $w=-1.09^{+0.09}_{-0.10}$. The highest-redshift data do not noticeably improve constraints for these models over the set of intermediate-redshift [SNe Ia]{} from ESSENCE+SNLS. It is for models with variable $w$ that the high-redshift data summarized by @riess04b provide the most utility. We here provide the global constraints on models characterized by $w=w_0+w_a(1-a)$ [@linder03; @albrecht06]. Using the BAO constraints on variable $w$ models would require integration from $z=0.35$ to $z\sim1089$ and the corresponding assumption that $w=w_0+w_a(1-a)$ is the proper parameterization over this stretch. If one is willing to make this assumption, then BAO+CMB already places significant constraints on the allowed $(w_0,w_a)$ parameter space. However, given that our multi-variable parameterizations of $w$ are arbitrary models with no clear theoretical motivation, we instead choose to regard $w=w_0+w_a(1-a)$ as a local expansion valid out to a redshift of $\sim2$ but not necessarily to $z\sim1089$. We then explicitly assume [${\Omega}_{\rm M}$]{}$=0.27\pm0.03$. Fig. \[fig:w0wa\_joint\_riess04\] shows the $(w_0, w_a)$ contours for this combined analysis. These constraints represent the advances of our understanding of dark energy. It is clear that work remains to constrain models of variable $w$.

Conclusions {#sec:conclusions}
===========

The ESSENCE survey has successfully discovered, confirmed, and followed 119 [SNe Ia]{} in our first four years of operation. We presented results from an analysis of [60]{} of those [SNe Ia]{} here, chosen so as to maximize insight while minimizing susceptibility to systematic errors. We have expended considerable effort to make quantitative estimates of various sources of systematic uncertainty that may afflict the ESSENCE results; of these, host-galaxy extinction and a potential local velocity monopole are currently the predominant concerns. We are working to devise ways to better estimate extinction, using both spectroscopic and photometric observations. Ideally, we would use all available information to arrive at an extinction prior customized for each supernova (e.g., different priors for elliptical and spiral host galaxies), rather than applying a single prior to the collection of all light curves.

The ESSENCE photometric calibration uncertainties will be reduced by an intensive calibration campaign this fall on the CTIO 4-m telescope in conjunction with the improved calibration of the SDSS southern stripe from the SDSS II project [@frieman04; @dilday05]. We hope to reduce our overall systematic uncertainty to the 5% level through this improved photometric calibration and an improved external nearby [SN Ia]{} sample from KAIT, the Nearby Supernova Factory, CfA, SDSS II, and the Carnegie SN Program to reduce our systematic sensitivity to a potential velocity monopole in the local [SN Ia]{} sample.

Combining our [SN Ia]{} observations with the BAO results of @eisenstein05 we find that a fit to a constant-$w$, flat-Universe model to our data finds an estimated parameter value of $w=$[$-1.05^{+0.13}_{-0.12}~{\rm (stat}~1\sigma{)} \pm 0.13~{\rm (sys)}$]{} with a [$\chi^2/{\rm DoF}$]{}$=$[$0.96$]{} using our full set analyzed with the MLCS2k2 fitter of @jha06c. A $w=-1$, flat-Universe model is consistent with our data. A combined analysis of ESSENCE+SNLS+nearby results in a estimated mean parameter of $w=$[$-1.07^{+0.09}_{-0.09}~{\rm (stat}~1\sigma{)} \pm 0.13~{\rm (sys)}$]{}. We have no reliable estimate of the systematic effects involving the SALT fitter but take our general systematic uncertainty of [$0.13$]{} as representative of the issues currently confronting supernova cosmology.

The statistical increase from the [SNe Ia]{} from the entire 6-year ESSENCE data set plus improved photometric calibration of our detector and photometric measurements will reduce our statistical uncertainty to 7% and, together with an improvement in our systematic uncertainties to the level 5%, allow us to surpass our goal of a 10% measurement of a constant $w$ in a flat Universe.

Establishing the nature of dark energy is among the most pressing issues in the physical sciences today. The emerging impression that the equation-of-state parameter is close to $w=-1$ makes it difficult to determine whether the underlying physics arises in the particle physics sector or from the classical cosmological constant of general relativity. A value of $w=-1$ is perhaps the least informative possible outcome. In our view, this state of affairs motivates a vigorous effort to push the observational constraints to improve our sensitivity to the value and derivative of $w$ and strongly encourages searching for other indications of new physics, as we well may need another piece to solve the puzzle handed us by Nature.

Acknowledgments
===============

Based in part on observations obtained at the Cerro Tololo Inter-American Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc. (AURA) under cooperative agreement with the National Science Foundation (NSF); the European Southern Observatory, Chile (ESO Programmes 170.A-0519 and 176.A-0319); the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the NSF (United States), the Particle Physics and Astronomy Research Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), CNPq (Brazil), and CONICET (Argentina) (Programs GN-2002B-Q-14, GS-2003B-Q-11, GN-2003B-Q-14, GS-2004B-Q-4, GN-2004B-Q-6, GS-2005B-Q-31, GN-2005B-Q-35); the Magellan Telescopes at Las Campanas Observatory; the MMT Observatory, a joint facility of the Smithsonian Institution and the University of Arizona; and the F. L. Whipple Observatory, which is operated by the Smithsonian Astrophysical Observatory. Some of the data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration; the Observatory was made possible by the generous financial support of the W. M. Keck Foundation.

The ESSENCE survey team is very grateful to the scientific and technical staff at the observatories we have been privileged to use.

[*Facilities:*]{} , , , , , , , , .

The survey is supported by the US National Science Foundation through grants AST-0443378, AST-057475, and AST-0607485. The Dark Cosmology Centre is funded by the Danish National Research Foundation. SJ thanks the Stanford Linear Accelerator Center for support via a Panofsky Fellowship. AR thanks the NOAO Goldberg fellowship program for its support. PMG is supported in part by NASA Long-Term Astrophysics Grant NAG5-9364 and NASA/HST Grant GO-09860. RPK enjoy support from AST06-06772 and PHY99-07949 to the Kavli Institute for Theoretical Physics. AC acknoledges the support of CONICYT, Chile, under grants FONDECYT 1051061 and FONDAP Center for Astrophysics 15010003.

Our project was made possible by the survey program administered by NOAO, and builds upon the data reduction pipeline developed by the SuperMacho collaboration. IRAF is distributed by the National Optical Astronomy Observatory, which is operated by AURA under cooperative agreement with the NSF. The data analysis in this paper has made extensive use of the Hydra computer cluster run by the Computation Facility at the Harvard-Smithsonian Center for Astrophysics. We also acknowledge the support of Harvard University.

This paper is dedicated to the memory of our friend and colleague Bob Schommer.

[^1]: <http://www.ctio.noao.edu/essence/>

[^2]: <http://snls.in2p3.fr/conf/release/>

[^3]: <http://www.ctio.noao.edu/essence/>

[^4]: <http://snls.in2p3.fr/conf/release/>

[^5]: <http://qold.astro.utoronto.ca/conley/simple_cosfitter/>

[^6]: <http://www.ctio.noao.edu/essence/>
