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THE ASTROPHYSICAL JOURNAL, 509 : 74 õ79, 1998 December 10
( 1998. The American Astronomical Society. All rights reserved. Printed in U.S.A.

SUPERNOVA LIMITS ON THE COSMIC EQUATION OF STATE PETER M. GARNAVICH,1 SAURABH JHA,1 PETER CHALLIS,1 ALEJANDRO CLOCCHIATTI,2 ALAN DIERCKS,3 ALEXEI V. FILIPPENKO,4 RON L. GILLILAND,5 CRAIG J. HOGAN,3 ROBERT P. KIRSHNER,1 BRUNO LEIBUNDGUT,6 M. M. PHILLIPS,7 DAVID REISS,3 ADAM G. RIESS,4 BRIAN P. SCHMIDT,8 ROBERT A. SCHOMMER,7 R. CHRIS SMITH,9 JASON SPYROMILIO,6 CHRIS STUBBS,3 NICHOLAS B. SUNTZEFF,7 JOHN TONRY,10 AND SEAN M. CARROLL11
Received 1998 May 28 ; accepted 1998 July 10

ABSTRACT We use Type Ia supernovae studied by the High-z Supernova Search Team to constrain the properties of an energy component that may have contributed to accelerating the cosmic expansion. We ïnd that for a ÿat geometry the equation-of-state parameter for the unknown component, a \ P /o , must be less x xx than [0.55 (95% conïdence) for any value of ) , and it is further limited to a \ [0.60 (95% m These values are inconsistent with the unknown x conïdence) if ) is assumed to be greater than 0.1. m topological defects such as domain walls, strings, or textures. The supernova (SN) data component being are consistent with a cosmological constant (a \[1) or a scalar ïeld that has had, on average, an x equation-of-state parameter similar to the cosmological constant value of [1 over the redshift range of z B 1 to the present. SN and cosmic microwave background observations give complementary constraints on the densities of matter and the unknown component. If only matter and vacuum energy are considered, then the current combined data sets provide direct evidence for a spatially ÿat universe with ) \ ) ] ) \ 0.94 ^ 0.26 (1 p). tot m " Subject headings : cosmology : observations õ cosmology : theory õ supernovae : general
1

. INTRODUCTION

Matter that clusters on the scale of galaxies or galaxy clusters is insufficient to close the universe, with conventional values near ) \ 0.2 ^ 0.1 (Gott et al. 1974 ; m Carlberg et al. 1996 ; Lin et al. 1996 ; Bahcall, Fan, & Cen 1997). Observations of distant supernovae (SNs) provide credible evidence that the deceleration rate of the universal expansion is small, which implies that the total matter density, clustered or smooth, is insufficient to create a ÿat geometry (Garnavich et al. 1998 ; Perlmutter et al. 1998). Either the universe has an open geometry or, if ÿat, other forms of energy are more important than matter. Large samples of SNs analyzed by the High-z Supernova Search collaboration (Riess et al. 1998a, hereafter R98a) and the Supernova Cosmology Project (Kim 1998) now suggest that the universe may well be accelerating. Matter alone
1 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138. 2 Departmento de Astronomia y Astrophisica, Pontiïcia Universidad Catolica, Casilla 104, Santiago 22, Chile. 3 Department of Astronomy, University of Washington, Seattle, WA 98195. 4 Department of Astronomy, University of California, Berkeley, CA 94720-3411. 5 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218. 6 European Southern Observatory, Karl-Schwarzschild-Strasse 2, Garching, Germany. 7 Cerro Tololo Inter-American Observatory, Casilla 603, La Serena, Chile. 8 Mount Stromlo and Siding Spring Observatory, Private Bag, Weston Creek P.O., Australia. 9 University of Michigan, Department of Astronomy, 834 Dennison, Ann Arbor, MI 48109. 10 Institute for Astronomy, University of Hawaii, Manoa, HI 96822. 11 Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106.

cannot accelerate the expansion ; therefore, if taken at face value the observations demand an additional energy component for the universe. While the vigorous pursuit of possible systematic eects (see, e.g., Hoÿich, Wheeler, & Thielemann 1998) will be important in understanding these observations, it is instructive to see what they imply about the energy content of the universe. The cosmological constant was revived to ïll the gap between the observed mass density and the theoretical preference for a ÿat universe (Turner, Steigman, & Krauss 1984 ; Peebles 1984), as well as to alleviate the embarrassment of a young universe with older stars (Carroll, Press, & Turner 1992). The cosmological constant is a negative pressure component arising from nonzero vacuum energy (Weinberg 1989). It would be extraordinarily difficult to detect on a small scale, but ) \ 1 [ ) could make up the dierence " between the matter densitym ) and a ÿat geometry and m might be detected by measurements on a cosmological scale. There are few independent observational constraints on the cosmological constant, but Falco, Kochanek, & Munoz (1998) estimated that ) \ 0.7 (95% conïdence) 8 " from the current statistics of strong gravitational lenses. If the matter density is less than ) D 0.3, this limit is close to m preventing the cosmological constant from making a ÿat geometry. Further, a cosmological constant that just happens to be of the same order as the matter content at the present epoch raises the issue of "" ïne tuning îî (Coles & Ellis 1997). A number of exotic forms of matter that might contribute to cosmic acceleration are physically possible and viable alternatives to the cosmological constant (Frieman & Waga 1998 ; Caldwell, Dave, & Steinhardt 1998). The range of possibilities can be narrowed by using SNs because the luminosity distance not only depends on the present densities of the various energy components but also depends on their equations of state while the photons we see were in ÿight. Here, with some simplifying assump74


SN LIMITS ON THE COSMIC EQUATION OF STATE tions, we consider the constraints that recent SN observations place on the properties of an energy component accelerating the cosmic expansion.
2

75

. OBSERVATIONS

The Type Ia SNs have been analyzed by the High-z Supernova Search Team and described by R98a, Garnavich et al. (1998), Schmidt et al. (1998), and Riess et al. (1998b). The full sample from R98a consists of 50 Type Ia SNs. Of these, 34 are at z \ 0.2, while the remaining 16 cover a range in redshift of 0.3 \ z \ 1.0. Six of the high-z events were analyzed using the "" snapshot îî method developed by Riess et al. (1998b). This innovative technique uses highquality spectra to deduce information unavailable because of a poorly sampled light curve. While the errors estimated from the snapshot method are larger than those from direct light-curve ïtting, the snapshot sample provides a signiïcant, independent set of Type Ia SNs distances. As shown by Phillips (1993), the light-curve decline rate of Type Ia SNs is correlated with the luminosity at maximum brightness of these exploding white dwarfs. This correlation has been calibrated by Hamuy et al. (1996 ; the *m (B) method) and by Riess, Press, & Kirshner (1995, 15 1996 ; the multicolor light curve shape, or MLCS, method that includes a correction for extinction), and both show that applying this correction to the Type Ia SNs Hubble diagram signiïcantly reduces the scatter. Phillips et al. (1998) extended the *m (B) approach to include an esti15 mate of the extinction. In R98a, an improved version of the MLCS method is presented. Here, as in R98a, we apply both MLCS and *m (B) (with extinction correction) tech15 niques to the analysis to gauge the systematic errors introduced by dierent light-curve ïtting methods.
3

. ANALYSIS

The apparent brightness of a Type Ia SN corrected for light-curve decline rate and extinction provides an estimate of the luminosity distance, D , from the K-corrected observed magnitude, m \ M ]L5 log D ] 25, and the absolute magnitude, M, of Type Ia SNs. L described by As Schmidt et al. (1998) and Carroll, Press, & Turner (1992), the luminosity distance depends on the content and geometry of the universe in a Friedmann-RobertsonWalker cosmology : c(1 ] z) D\ sinn L H Jo ) o 0 k z ~1@2 ; ) (1 ] z@)3(1`ai)] ) (1 ] z@)2 dz@ , ] Jo ) o k i k 0i (1) sinh (x) , if ) [ 0 ; k if ) \ 0 ; (2) sinn(x) \ x , k sin (x) , if ) \ 0 , k where ) are the normalized densities of the various energy i components of the universe and ) \ 1 [ ; ) describes k ii the eects of curvature. The exponent n \ 3(1 ] a) deïnes the way each component density varies as the universe expands, o P a~n, where a is the cosmic scale factor. For example, n has the value 3 for normal matter since the mass density declines proportionally to the volume. Alternative-

G

PC

D

H

7

ly, a is the equation-of-state parameter for component i i deïned as the ratio of the pressure to the energy density, a \ P /o (sometimes denoted in the literature as w). The i ii relation n \ 3(1 ] a) is easily derived from the conservation-of-energy equation in comoving coordinates (see, e.g., eq. [15.1.21] of Weinberg 1972). In the most general case, the equation of state can vary with time in ways other than assumed here (as the sum of power laws in 1 ] z), but we are limited by the quality and range of the SN observations to consider only its average eect between the present and z \ 1. The present-day value of the Hubble constant (H ) and the absolute magnitude of Type Ia SNs 0 (M) are primarily set by the low-z sample, which allows the high-z events to constrain the cosmological eects. This means that conclusions derived from Type Ia SN are independent of the absolute distance scale. Gravitational lensing by matter distributed between the observer and SNs at high z can aect the observed brightness of Type Ia SNs and induce errors in the estimate of their luminosity distances (Kantowski, Vaughan, & Branch 1995). For realistic models of the matter distribution and ) \ 0.5, the most likely eect of the lensing is to make the m SNs at z \ 0.5 about 2% fainter than they would appear if the matter were distributed uniformly (ïlled beam) as shown by Wambsganss et al. (1997). Holz & Wald (1998) have shown that the magnitude of the eect also depends on whether the matter is distributed smoothly on galaxy scales or is clumped in MACHOS, but the error induced remains small when ) \ 0.5. For simplicity, our calculations conm sider only the ïlled-beam case ; however, the eect of assuming the extreme case of an empty beam is shown by Holz (1998). There are a few known, and possibly some unknown, energy components that aect D . Ordinary gravitating L matter, ) , certainly has had some eect on the universal mbetween z B 1 and now. Since the matter density expansion scales inversely with the volume, a \ 0, and matter (baryons, neutrinos, and dark matter ; m formerly earth, air, and water) contributes no pressure. Radiation (ïre in an earlier lexicon) (a \ 1 ) dominated during a period in the 3 early universe butris negligible for z \ 1. Equation (1) shows that for nonÿat models the curvature term ) contributes to k the luminosity distance like a component with a \[ 1 , but k 3 additional geometrical eects as prescribed by equation (2) are also important. Other more speculative components have been proposed. A nonzero vacuum energy, ) , is a popular possibility " explored by R98a for this data set. Because the vacuum energy density remains constant as the universe expands (that is, o P a0), we have a \[1. Topological defects " created in " early universe could also leave remnants that the might contribute to the energy now. Networks of cosmic strings may be a natural consequence of phase transitions in the young universe, and if they did not intercommute they would have an average eective a \[ 1 (Vilenkin 1984 ; 3 Spergel & Pen 1997). A network of scomoving domain walls would have an average equation-of-state parameter of [ 2 3 (Vilenkin 1985), while a globally wound texture would produce a \[ 1 (Davis 1987 ; Kamionkowski & Toumbas t 3 1996). Evolving cosmic scalar ïelds with suitable potentials could produce a variety of exotic equations of state with signiïcant densities at the present epoch (Peebles & Ratra 1988 ; Frieman et al. 1995 ; Frieman & Waga 1998). Scalar


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ïelds could also produce variable mass particles (VAMPs) that would redshift more slowly than ordinary matter, which would create an eective a \ 0 (Anderson & VAMP Carroll 1998). These ïelds may evolve over time and would produce an interesting variety of cosmic histories. Our goal is modest : we only hope to constrain the average a over the range where Type Ia SNs are presently observed. To simplify the analysis, we assume that only one component in addition to gravitating matter aects the cosmic expansion. Because the origin of the acceleration is unknown, we will refer to this as the x-component with a density of ) and an equation of state of P \ a o . x x xx Caldwell et al. (1998) have dubbed the unknown component "" quintessence îî since the other four essences have already been employed above. We assume that the universe on very large scales is accurately described by general relativity and that the x-component obeys the null energy condition (NEC). The NEC states that, for any null vector vk, the energy-momentum tensor satisïes T vkvl º 0 (see, e.g., Wald 1984). This is the weakest of all kl conventional energy conditions and should be satisïed by any classical source of energy and momentum including those discussed above. In a Robertson-Walker metric, the NEC is equivalent to requiring o ] P º 0. The NEC therefore restricts the x x energy density of the unknown component to be positive for a [ [1 and negative when a \ [1, while the energy x x density of the cosmological constant (a \[1) is unconx strained.

RESULTS

First, we ïx the equation of state of the unknown component and estimate the probability density function for the parameters ) , ) , and H given the observed Type Ia SNs xm 0 distance moduli. The joint likelihood distributions are then calculated in the same way as by R98a and are shown for representative values of a in Figure 1. Here we integrate x over all possible H with the prior assumption that all 0 values are equally likely. For a \ [0.7, the derived conx straints are similar to those found by R98a for a cosmological constant (a \[1) ; however, as the equation-ofx state parameter increases, the major axis of the uncertainty ellipses rotates about one point on the ) \ 0 line. For an x accelerating universe, the pivot point is on the negative ) m side. When a [ [0.4, the x-component could not reprox duce the observed acceleration, and all of these models give a poor ït to the observed Type Ia SN data ; the best ït occurs for a completely empty universe. Next, we allow the equation of state to vary freely, but we restrict the densities to ) ] ) \ 1, or open models. We m x then integrate over all possible values of ) assuming a x uniform prior distribution to provide the joint probability for a and the matter density. From the NEC we must x include regions where a \ [1 and ) is negative, but these x are unable to produce x accelerating universe and therean fore they have a very low probability. For open models, highest joint probabilities are conïned to a region bounded

= -0.9

= -0.7

= -0.3 = -0.5

FIG. 1.õJoint probability distributions for ) and the density of the unknown component, ) , based on the Type Ia SN magnitudes reduced with the x MLCS method. Four representative values of them equation-of-state parameter, a , are shown. See R98a for the distribution when a \[1. x x


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OTHER CONSTRAINTS

MLCS+Snap

m 15 (B) +Snap

FIG. 2.õJoint probability distributions from Type Ia SNs for ) and m the equation-of-state parameter a assuming a ÿat spatial geometry x () ] ) \ 1). The top panel uses SN distances from the MLCS method m x combined with SNs reduced using the snapshot method, while the bottom panel is from the *m (B) technique plus snapshot results. The vertical 15 dotted line marks the matter density estimated from galaxy cluster dynamics.

by [1.0 \a \ [0.4 and ) \ 0.2. If we consider any x m value of ) equally likely, then a \ [0.47 for the MLCS m a \ [0.64 for the *m (B) results with 95% x method and x 15 conïdence. Finally, we consider ÿat models for the universe. The joint probability between the equation-of-state parameter and the matter density for ) ] ) \ 1 is shown in Figure x 2. The two cases are for the m MLCS and the *m (B) light15 curve ïts, and they demonstrate that the two methods for deriving luminosity from light curves provide consistent constraints. Note that in the ÿat case, the NEC allows a \ x [1 only when ) [ 1 has an insigniïcant probability and m plots can be compared to pioneering is not plotted. These calculations by Turner & White (1997) and White (1998), which used smaller SN samples. The improved Type Ia SNs data favor acceleration and support both a low ) and a m small value of a . Integrating the probability over all values xa uniform prior shows that a \ [0.55 for of ) assuming m x MLCS and a \ [0.63 for *m (B) (95% conïdence). If we x 0.1, then the limits tighten to a \ [0.60 15 assume ) [ m x (MLCS) and a \ [0.69 [*m (B)]. For matter densities x 15 near D0.2 favored by galaxy cluster dispersions, the most probable equation-of-state parameters are between [0.7 and [1.0. These results disfavor topological defect models such as domain walls (90% conïdence) and eliminate strings and textures (99% level) as the principal component of the unknown energy. The cosmological constant, or a form of quintessence that resembles it for z \ 1, is supported by the data. Constraints that refer to higher z are needed to narrow the range of possible models.

High-z Type Ia SNs observations combined with the cosmic microwave background (CMB) anisotropy angular power spectrum provide complementary constraints on the densities of matter and the x-component (White 1998 ; Tegmark et al. 1998). Details of the CMB power spectrum depend on a large number of variables, but the angular scale of the ïrst acoustic peak primarily depends on the physics of recombination and the angular diameter distance to the surface of last scattering (Kamionkowski, Spergel, & Sugiyama 1994 ; White & Scott 1996 ; White 1998). Rather than ït the power spectrum in detail, we have restricted our attention to the location of the ïrst acoustic peak as estimated from current CMB experiments (Hancock et al. 1998). This is a rapidly moving experimental ïeld, and new results will surely supersede these, but they illustrate the power of combining the SN data with the CMB. We employ the analytic approximations of White (1998) to determine the wavenumber of the acoustic peak at recombination, and those of Hu & Sugiyama (1996) to determine the recombination redshift ; thus we assume adiabatic ÿuctuations generate the anisotropy. In addition, we have ignored reionization and ïxed the number of neutrino species at three, as well as assumed only scalar modes with a spectral index of n \ 1. A thorough treatment of this problem would allow all of these parameters to vary and integrate the probability over all possible values (that is, marginalize over them) ; however, this would be very time consuming, even with the fast CMB code of Seljak & Zaldarriaga (1996), and disproportionate to the precision of the current data. A large exploration of the parameter space involved (though lacking a full variation of ) ) can be found in Bartlett et al. " (1998) and Lineweaver (1998). Our calculation determines the angular scale multipole of the ïrst acoustic peak for a grid in a three-dimensional parameter space of () , ) , H ), where we explicitly allow M 0 for open, ÿat, and closed" universes with and without a cosmological constant. We also employ the additional constraint on the baryon density ) h2\ 0.024 (h \ H /100 km b 0 s~1 Mpc~1) derived from the primordial deuterium abundance and nucleosynthesis (Tytler, Fan, & Burles 1996). Other estimates of the baryon fraction (see Fugikita, Hogan, & Peebles 1998) could be used, but the location of the peak depends only weakly on this parameter. Where possible, we checked these calculations with numerical integrations (Seljak & Zaldarriaga 1996) and conïrmed that the peak locations agree to [10%, which is adequate for this exploration. Following White (1998), we combined the predicted peak location with the observations using a phenomenological model for the peak (Scott, Silk, & White 1995). Recent CMB measurements analyzed by Hancock et al. (1998) give the conditional likelihood of the ïrst acoustic peak position as l \ 263`139 , based on best-ït values of the peak peak ~94 amplitude and low-multipole normalization. Rocha et al. (1998) have provided us with a probability distribution function for the ïrst peak position based on marginalizations over the amplitude and normalization that is a more general approach than that by Hancock et al. (1998). The Rocha et al. (1998) function gives l \ 284`191 , peak ~84 which is only a small shift from the value derived using the conditional likelihood method. We then marginalize the likelihood in our three-dimensional parameter space over


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H with a Gaussian prior based on our own Type Ia SNs 0 result including our estimate of the systematic error from the Cepheid distance scale, H \ 65 ^ 7 km s~1 Mpc~1 0 (R98a). It is important to note that the Type Ia SNs constraints on () , ) ) are independent of the distance scale m" but that the CMB constraints are not. We then combine marginalized likelihood functions of the CMB and Type Ia SNs data. The result is shown in Figure 3. Again, we must caution that systematic errors in either the Type Ia SNs data (R98a) or the CMB could aect this result. Nevertheless, it is heartening to see that the combined constraint favors a location in this parameter space that has not been ruled out by other observations, though there may be mild conÿict with constraints on ) from gravitational " lensing (Falco et al. 1998). In fact, the region selected by the Type Ia SN and CMB observations is in concordance with inÿation, large-scale structure measurements, and the ages of stars (Ostriker & Steinhardt 1995 ; Krauss & Turner 1995). The combined constraint removes much of the high) , high-) region that was not ruled out by the Type Ia m " SN data alone, as well as much of the high-) , low-) m " region allowed by the CMB data alone. The combined constraint is consistent with a ÿat universe, as ) \ ) tot m ] ) \ 0.94 ^ 0.26 for MLCS and 1.00 ^ 0.22 for *m (B) " 15the (1 p errors). The enormous redshift dierence between CMB and the Type Ia SNs makes it dangerous to generalize

this result beyond a cosmological-constant model because of the possible time dependence of a . But for an equation x of state ïxed after recombination, the combined constraints continue to be consistent with a ÿat geometry as long as a [ [0.6. With better estimates of the systematic errors in x the Type Ia SN data and new measurements of the CMB anisotropy, these preliminary indications should quickly turn into very strong constraints (Tegmark et al. 1998).
6

. CONCLUSIONS

The current results from the High-z Supernova Search Team suggest that there is an additional energy component sharing the universe with gravitating matter. For a ÿat geometry, the ratio of the pressure of the unknown energy to its density is probably more negative than [0.6. This eectively rules out topological defects such as strings and textures as the additional component, and it disfavors domain walls as that component. Open models are less constrained but favor a \ [0.5. Although there are many x intriguing candidates for the x-component, the current Type Ia SN observations imply that a vacuum energy or a scalar ïeld that resembles the cosmological constant is the most likely culprit. Combining the Type Ia SNs probability distribution with constraints of today from the position of the ïrst acoustic peak in the CMB power spectrum provides a simultaneous

FIG. 3.õCombined constraints from Type Ia SNs and the position of the ïrst Doppler peak of the CMB angular power spectrum. The equation-of-state parameter for the unknown component is like that for a cosmological constant, a \[1. The contours mark the 68%, 95.4%, and 99.7% enclosed x probability regions.


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observational measurement of the densities of matter and of the unknown component. Using CMB data from Hancock et al. (1998) and following the analysis by White (1998), the result favors a ÿat universe with ) \ 0.94 ^ 0.26, domitot nated by the x-component for a B [1. Given the rapid x improvement in both the study of Type Ia SNs and the CMB, we can expect more powerful inferences about the contents of the universe to follow. We thank U. Seljak and M. Tegmark for some informative discussions, and G. Rocha for making the CMB likeli-

hood function available before publication. This work was supported by grant GO-7505 from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555, and at Harvard University through NSF grants AST 92-21648 and AST 95-28899, and an NSF Graduate Research Fellowship. Work at the University of Washington was supported through NSF grant AST 9617036. A. V. F. acknowledges the support of NSF grant AST 94-17213. S. M. C. was supported by NSF grant PHY/ 94-07194.

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