Supernova Ia Constraints on a Time-Variable Cosmological

Home , Bharat Ratra

SUPERNOVA Ia CONSTRAINTS ON A TIME-VARIABLE COSMOLOGICAL ““ CONSTANT ÏÏ SILVIU PODARIU AND BHARAT RATRA Department of Physics, Kansas State University, Manhattan, KS 66506 Received 1999 September 2; accepted 1999 November 3

ABSTRACT The energy density of a scalar Ðeld / with potential V (/) P /~a, a[0, behaves like a time-variable cosmological constant that could contribute signiÐcantly to the present energy density. Predictions of this spatially Ñat model are compared to recent Type Ia supernova apparent magnitude versus redshift data. A large region of model parameter space is consistent with current observations. (These constraints are based on the exact scalar Ðeld model equations of motion, not on the widely used time-independent equation of state Ñuid approximation equations of motion.) We examine the consequences of also incorporating constraints from recent measurements of the Hubble parameter and the age of the universe in the constant and time-variable cosmological constant models. We also study the e†ect of using a noninformative prior for the density parameter. Subject headings: cosmology: observations È large-scale structure of universe È supernovae: general

1. INTRODUCTION 4. The need for mild antibiasing to accommodate the excessive intermediate- and small-scale power predicted in Current observations are more consistent with lower- the COBE DMRÈnormalized Ñat constant-" CDM model density cosmogonies dominated by cold dark matter with a scale-invariant spectrum (see, e.g., Cole et al. 1997). (CDM). For recent discussions see, e.g., Ratra et al. (1999a), Kravtsov & Klypin (1999), Doroshkevich et al. (1999), While the time-variable " model has not yet been studied Colley et al. (2000), Freudling et al. (1999), Sahni & Star- to the same extent as the open and Ñat-" cases, it is likely obinsky (1999), Bahcall et al. (1999), and Donahue & Voit that it can be reconciled with most of these observations (1999). The simplest low-density CDM models have either (see, e.g., PR; Ratra & Quillen 1992, hereafter RQ; Frieman Ñat spatial hypersurfaces and a constant or time-variable & Waga 1998; Perlmutter, Turner, & White 1999b; Wang cosmological ““ constant ÏÏ " (see, e.g., Peebles 1984; Peebles et al. 2000; Efstathiou 1999). & Ratra 1988, hereafter PR; Efstathiou, Sutherland, & We emphasize that most of these observational indica- Maddox 1990; Stompor,Go rski, & Banday 1995; Caldwell, tions are tentative and certainly not deÐnitive. This is par- 2 Dave, & Steinhardt 1998) or open spatial hypersurfaces and ticularly true for constraints derived from a s comparison no " (see, e.g., Gott 1982, 1997; Ratra & Peebles 1994, between model predictions and cosmic microwave back- 1995;Go rski et al. 1998). ground (CMB) anisotropy measurements (see, e.g., Ganga, While these models are consistent with most recent Ratra, & Sugiyama 1996; Lineweaver & Barbosa 1998; observations, there are notable exceptions. For instance, Baker et al. 1999; Rocha 1999); see discussions in Bond, recent applications of the apparent magnitude versus red- Ja†e, & Knox (1998) and Ratra et al. (1999b). More reliable shift test based on Type Ia supernovae (SNe Ia) favor a constraints follow from model-based maximum likelihood nonzero " (see, e.g., Riess et al. 1998, hereafter R98; Perl- analyses of the CMB anisotropy data (see, e.g.,Go rski et al. mutter et al. 1999a, hereafter P99), although not with great 1995; Yamamoto & Bunn 1996; Ganga et al. 1998; Ratra et statistical signiÐcance (Drell, Loredo, & Wasserman 2000). al. 1999b; Rocha et al. 1999), which make use of all the On the other hand, the open model is favored by the information in the CMB anisotropy data and are based on following: fewer approximations. However, this technique has not yet been used to analyze enough data sets to provide robust \ 1. Measurements of the Hubble parameterH0 ( 100 h statistical constraints. Kamionkowski & Kosowsky (1999) km s~1 Mpc~1) that indicate h \ 0.65 ^ 0.1 at 2 p (see, e.g., review what might be expected from the CMB anisotropy in Suntze† et al. 1999; Biggs et al. 1999; Madore et al. 1999), the near future. and measurements of the age of the universe that indicate In this paper we examine constraints on a constant and \ ^ t0 12 2.5 Gyr at 2 p (see, e.g., Reid 1997; Gratton et al. time-variable " that follow from recent Type Ia supernova 1997; Chaboyer et al. 1998). This is because the resulting apparent magnitude versus redshift data and recent mea- centralH0 t0 value is consistent with a low-density open surements ofH0 andt0. We focus here on the favored scalar model with nonrelativistic-matter density parameter )0 B Ðeld (/) model for a time-variable " (PR; Ratra & Peebles 0.35 and requires a rather large)0 B 0.6 in the Ñat 1988, hereafter RP), in which the scalar Ðeld potential constant-" case. V (/) P /~a, a[0, at low redshift.1 This scalar Ðeld could 2. Analyses of the rate of gravitational lensing of quasars and radio sources by foreground galaxies that require a 1 Other potentials have been considered, e.g., an exponential potential large)0 º 0.38 at 2 p in the Ñat constant-" model (see, e.g., (see, e.g., Lucchin & Matarrese 1985; RP; Ratra 1992; Wetterich 1995; Falco, Kochanek, & Munoz 1998). Ferreira & Joyce 1998), or one that gives rise to an ultralight pseudoÈ Nambu-Goldstone boson (see, e.g., Frieman et al. 1995; Frieman & Waga 3. Analyses of the number of large arcs formed by strong 1998), but such models are either inconsistent with observations or do not gravitational lensing by clusters (Bartelmann et al. 1998; share the more promising features of the inverse power law potential Meneghetti et al. 2000). model. 109 110 PODARIU & RATRA Vol. 532

FIG. 1.ÈPDF conÐdence contours for the spatially Ñat time-variable " scalar Ðeld model, with potential V (/) P /~a, derived using the three SN Ia data sets. The a \ 0 axis corresponds to the spatially Ñat time-independent " case. ConÐdence contours in panels aÈc run from [2to ] 3 p starting from the lower left-hand corner of each panel. Panel a shows those derived from all the R98 SNe, while panel b is for R98 SNe excluding the z \ 0.97 one, and panel c is for the P99 Ðt C data set. Panel d compares the ^2 p limits from the three data sets: all R98 SNe (long-dashed lines); R98 SNe excluding the z \ 0.97 one (short-dashed lines); and the P99 Ðt C SNe (dotted lines). have played the role of the inÑation at much higher redshift, lence may be used to show that a scalar Ðeld with potential with the potential V (/) dropping to a nonzero value at the V (/) P /~a, a[0, acts like a Ñuid with negative pressure end of inÑation and then decaying more slowly with and that the / energy density behaves like a cosmological increasing / (PR; RP). See Peebles & Vilenkin (1999), Per- constant that decreases with time. This energy density could rotta & Baccigalupi (1999), and Giovannini (1999) for a come to dominate at low redshift and thus help reconcile speciÐc model and observational consequences of this sce- low dynamical estimates of the mean mass density with a nario. A potential P/~a could arise in a number of high- spatially Ñat cosmological model. Alternative mechanisms energy particle physics models; see, e.g.,Bine truy (1999), that also rely on negative pressure to achieve this result Kim (1999), Barr (1999), Choi (1999), Banks, Dine, & have been discussed (see, e.g.,Ozer & Taha 1986; Freese et Nelson (1999), Brax & Martin (1999), Masiero, Pietroni, & al. 1987; Turner & White 1997; Chiba, Sugiyama, & Naka- Rosati (2000), and Bento & Bertolami (1999) for speciÐc mura 1997;O zer 1999; Waga & Miceli 1999; Overduin examples. It is conceivable that such a setting might provide 1999; Bucher & Spergel 1999). However, these mechanisms an explanation for the needed form of the potential, as well either are inconsistent or do not share a very appealing as for the needed very weak coupling of / to other Ðelds feature of the scalar Ðeld models. For some of the scalar (RP; Carroll 1998; Kolda & Lyth 1999; but see Periwal Ðeld potentials mentioned above, the solution of the equa- 1999 and Garriga, Livio, & Vilenkin 2000 for other possible tions of motion is an attractive time-dependent Ðxed point explanations for the needed present value of "). (RP; PR; Wetterich 1995; Ferreira & Joyce 1998; Copel- A scalar Ðeld is mathematically equivalent to a Ñuid with and, Liddle, & Wands 1998; Zlatev, Wang, & Steinhardt a time-dependent speed of sound (Ratra 1991). This equiva- 1999; Liddle & Scherrer 1999; Santiago & Silbergleit 1998; No. 1, 2000 SN Ia CONSTRAINTS ON COSMOLOGICAL ““ CONSTANT ÏÏ 111

FIG. 2.È(a) PDF conÐdence contours derived from the P99 Ðt C SNe, for a spatially Ñat Ñuid model with equation of state p \ wo (solid lines show contours from [3to]3 p starting from the lower left-hand corner) and for the spatially Ñat time-variable " scalar Ðeld model with potential V (/) P /~a, \ [ ] which behaves like a Ñuid model with equation of statepÕ wÕ oÕ in the CDM- and baryon-dominated epoch (dotted lines show contours from 2to 3 p starting from the lower left-hand corner). The scalar Ðeld model contours were derived by using eq. (1) to transform the vertical axis of Fig. 1c. The Ñuid model likelihood function was computed for the full range of w and)0 shown. The dot-dashed lines bound thewÕ È)0 region that corresponds to theaÈ)0 region of Fig. 1c over which the scalar Ðeld model likelihood was computed. (b) Contours of constantw (eq. [2]) in theaÈ) plane for the time-variable " P ~a eff \[ 0 [ [ [ [ [ scalar Ðeld model with potential V (/) / . Starting at the lower left-hand corner, the contours correspond to weff 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, [0.35, [0.3, and [0.25.

Uzan 1999). This means that for a wide range of initial supernova data of R98 and P99 to derive constraints on a conditions the scalar Ðeld / evolves in a manner that time-variable " without making use of the time- ensures that the cosmological constant contributes a rea- independent equation of state Ñuid approximation to the sonable energy density at low redshift. Of course, this does scalar Ðeld model. not resolve the (quantum-mechanical) cosmological con- In the analyses here we use the most recent R98 and P99 stant problem. data to place constraints on a constant and time-variable ". There have been many studies of the constraints placed We note, however, that this is a young Ðeld and insufficient on a time-variable " from Type Ia supernova apparent understanding of a number of astrophysical processes and magnitude versus redshift data (see, e.g., Turner & White e†ects (the mechanism responsible for the supernova, evolu- 1997; Frieman & Waga 1998; Garnavich et al. 1998; Hu et tion, environmental e†ects, intergalactic dust, etc.) could al. 1999; Cooray 1999; Waga & Miceli 1999; P99; Perlmut- render this a premature undertaking (see, e.g.,Ho Ñich et al. ter et al. 1999b; Wang et al. 2000; Efstathiou 1999). 1998; Aguirre 1999; Drell et al. 2000; Umeda et al. 1999; However, except for the analysis of Frieman & Waga Riess et al. 1999; Wang 2000). Furthermore, other cosmo- (1998), who use the earlier Perlmutter et al. (1997) data, logical explanations (time-variable gravitational or Ðne- these analyses have mostly made use of the time- structure ““ constants ÏÏ) could be consistent with the data independent equation of state Ñuid approximation to the (see, e.g., Amendola, Corasaniti, & Occhionero 1999; scalar Ðeld model for a time-variable ". (Perlmutter et al. Barrow & Magueijo 2000;Garc• a-Berro et al. 1999). 1999b and Efstathiou 1999 do go beyond the time- In addition to analyzing just the supernova data, we also independent equation of state approximation by also incorporate constraints from recent measurements of H0 approximating the time dependence of the equation of andt0 and derive a combined likelihood function which we state; however, they do not analyze the exact scalar Ðeld use to constrain both a constant and a time-variable ".We model.)2 A major purpose of this paper is to use the newer also examine the e†ect on the model-parameter constraints caused by varying the prior for ) . 2 A similar criticism holds for most analyses of the constraints on a 0 time-variable " from gravitational lensing statistics (see, e.g., BloomÐeld We emphasize that the tests examined in this paper are Torres & Waga 1996; Jain et al. 1998; Cooray 1999; Waga & Miceli 1999; not sensitive to the spectrum of inhomogeneities in the Wang et al. 2000), with the exception of the analysis of RQ. models considered. This spectrum (possibly generated by 112 PODARIU & RATRA Vol. 532

FIG. 3.ÈPDF conÐdence contours derived from the P99 Ðt C SNe, for the spatially Ñat time-variable " scalar Ðeld model with potential V (/) P /~a. The dotted lines in panels aÈc show the [2to]3 p contours of Fig. 1c. The solid lines in these panels show the corresponding contours derived from the SN data in conjunction with: (a)H measurements (using eq. [3]); (b)t measurements (using eq. [4]); and (c)H andt measurements (using eqs. [3] and [4]). Panel ^ 0 0 0 0 d compares the 2 p conÐdence contours from the SN data in conjunction with: theH0 constraint (long-dashed lines), thet0 constraint (short-dashed lines), and both constraints (dotted lines). quantum-mechanical zero-point Ñuctuations during inÑa- intergalactic dust has a negligible e†ect (Aguirre 1999 con- tion; see, e.g., Fischler, Ratra, & Susskind 1985) is relevant siders the e†ects of dust). for some of the other tests mentioned above (CMB anisot- Our analysis of the data of R98 is similar to that ropy, antibiasing, etc.). described in their ° 4, with a few minor modiÐcations. For In ° 2 we summarize the computations. Results are pre- the time-independent " model we compute the likelihood sented and discussed in ° 3. Conclusions are given in ° 4. functionL ()0, )", H0) for a range of)0 spanning the interval 0È2.5 in steps of 0.1, for a range of) spanning the [ " 2. COMPUTATION interval 1 to 3 in steps of 0.1, and for a range of H0 spanning the interval 50È80 km s~1 Mpc~1 in steps of 0.5 We examine three sets of supernova apparent magni- km s~1 Mpc~1. tudes. We use the MLCS magnitudes for the R98 data, both Our analysis of the P99 data is similar to theirs, with including and excluding the unclassiÐed SN 1997ck at some modiÐcations. SpeciÐcally, we work with their z \ 0.97 (with 50 and 49 SNe Ia, respectively). The third set corrected/e†ective magnitudes and so need only compute a are the corrected/e†ective stretch factor magnitudes for the three-dimensional likelihood functionL ()0, )", MB). Here 54 Ðt C SNe Ia of P99. In all three cases we assume that the MB is their ““ Hubble constantÈfree ÏÏ B-band absolute mag- measured magnitudes are independent. We also assume nitude (see P99) related toH by M \[19.46 [ ] 0 B that SNe Ia at high and low z are not signiÐcantly di†erent 5log10 H0 25 (determined by us from results in P99). (Drell et al. 2000, Riess et al. 1999, and Wang 2000 consider For the time-independent " model we compute this likeli- the possibility and consequences of evolution), and that hood function for a range of)0 spanning the interval 0È3in No. 1, 2000 SN Ia CONSTRAINTS ON COSMOLOGICAL ““ CONSTANT ÏÏ 113 [ \ ^ steps of 0.1, for a range of)" spanning the interval 1.5 to and from measurements that indicatet0 12 1.3 Gyr at 3 in steps of 0.1, and for a range ofMB spanning the interval 1 p (i.e., 0.5 Gyr added to the globular cluster age estimate [3.95 to [2.95 in steps of 0.05 (which corresponds to of Chaboyer et al. 1998),3 by using the prior about the same interval inH0 used in our analyses of the R98 data). Note that in our P99 time-independent " model \ 1 p(t0) plots we do not show the likelihood for the whole )0 È)" J2nM1.3 GyrN region over which we have computed it. " Mt (H , ) , ) ) [ 12 GyrN2 The spatially Ñat time-variable model we consider is ] exp C[ 0 0 0 " D , (4) the scalar Ðeld model with potential V (/) P /~a, a[0. 2M1.3 GyrN2 When a ] 0 the model tends to the Ñat constant-" case and when a ] O it approaches the EinsteinÈde Sitter model. It with a replacing)" in this expression in the spatially Ñat is discussed in detail in PR, RP, and RQ, and we derive the time-variable " case. Note that Figure 2 of Chaboyer et al. predicted distance moduli for the SNe using expressions (1998) may be used to establish that a Gaussian prior of the given in these papers. form of equation (4) is a good approximation to the shifted As shown in these papers, the scalar Ðeld behaves like a Monte Carlo globular cluster age distribution. See Ganga Ñuid with a constant (but di†erent) equation of state in each et al. (1997) for a discussion of Gaussian priors in a related epoch of the model. For instance, in the CDM- and baryon- context. This method of incorporating constraints from dominated epoch of the model, it obeys the equation of measurements ofH0 andt0 is not identical to the classical \ H t cosmological test (see, e.g., ° 13 of Peebles 1993). statepÕ wÕ oÕ (relating the pressure and energy density of 0 0 the scalar Ðeld), where Finally, since)0 is a positive quantity, we also consider the noninformative prior (Berger 1985, p. 82),4 \[ 2 wÕ ; (1) 1 a ] 2 p() ) \ . (5) 0 ) see, e.g., equation (2) of RQ, or see Zlatev et al. (1999) for a 0 more recent derivation. We shall also have need for an Since)0 is bounded from below by the observed lower limit average equation-of-state parameter, used by Perlmutter et on the baryon density parameter (D0.01È0.05, depending al. (1999b) and Wang et al. (2000), on the data used; see, e.g., Olive, Steigman, & Walker 2000), this prior does not result in an inÐnity. /a0 da ) (a)w (a) w \ 0 Õ Õ , (2) eff a0 RESULTS AND DISCUSSION /0 da )Õ(a) 3. where)Õ is the scalar Ðeld density parameter and a is the Figure 1 shows the posterior probability density distribu- scale factor (witha0 being the present value). tion function (PDF) conÐdence contours for the time- In this model, for the R98 data we compute the likelihood variable " scalar Ðeld model, derived using the three data functionL ()0, a, H0), and for the P99 data the likelihood sets discussed above. Figure 1d shows that constraints from functionL ()0, a, MB). In both cases we evaluate the likeli- the three di†erent data sets are quite consistent. At 2 p a hood function for a range of)0 spanning the interval 0.05È large region of the parameter space of these spatially Ñat 0.95 in steps of 0.025, for a range of a spanning the interval models (with a constant or time-variable ") is consistent 0È8 in steps of 0.5, and for the same range ofH0 orMB as in with the SN Ia data. These data favor a smaller)0 as a is the time-independent " cases discussed above. increased from zero. We marginalize these three-dimensional likelihood func- A Ñuid with a time-independent equation of state p \ wo, tions by integrating overH0 (for the R98 data) orMB (for w \ 0, has often been used to approximate the scalar Ðeld the P99 data) and derive two-dimensional likelihood func- with potential V (/) P /~a in the time-variable " model. tions,L ()0, )") for the constant-" model andL ()0, a) for The solid lines in Figure 2a show the conÐdence contours the spatially Ñat time-variable " case. These two- for such a Ñuid model, derived using the P99 Ðt C data. dimensional likelihood functions are used to derive highest These conÐdence contours are consistent with those shown posterior density limits (see Ganga et al. 1997 and references in Figure 1 of Perlmutter et al. (1999b) (but see discussion therein) in the()0, )")( or)0, a) planes. In what follows we below), Figure 10 of Wang et al. (2000), and Figure 4 of consider 1, 2, and 3 p conÐdence limits which include Efstathiou (1999). The dashed lines in Figure 2a show the 68.27%, 95.45%, and 99.73% of the area under the likeli- exact scalar Ðeld model contours of Figure 1c, transformed hood function. using the relation betweenwÕ and a in the CDM- and When marginalizing over a parameter or deriving a limit baryon-dominated epoch (eq. [1]), which comes to an end from the likelihood functions, we consider a number of dif- just before the present. The two sets of contours agree near ferent priors. We Ðrst consider a uniform prior in the w D [1, as they must, since at w \[1 this is the Ñat parameter integrated over, set to zero outside the range constant-" model and " does behave exactly like a Ñuid considered for the parameter. We also incorporate constraints from measurements that 3 indicateH \ 65 ^ 7 km s~1 Mpc~1 at 1 p (see, e.g., Biggs A more complete analysis would need to account for the uncertainty 0 in this (0.5 Gyr) numerical value. This would likely weaken the e†ect of this et al. 1999; Madore et al. 1999), by using the prior prior. 4 We thank R. Gott for emphasizing this prior; a more complete dis- \ 1 cussion of it may be found in J. R. Gott et al. (2000, in preparation). p(H0) J2nM7kms~1 Mpc~1N I. Wasserman has noted that such a prior is probably more appropriate for a parameter that sets the scale for the problem, such asH0 here (see ° B.2 MH [ 65 km s~1 Mpc~1N2 of Drell et al. 2000; ° VII of Jaynes 1968 gives a more general discussion). It ] exp C[ 0 D , (3) is, however, still of interest to determine how the conclusions depend on 2M7kms~1 Mpc~1N2 the choice of prior. 114 PODARIU & RATRA Vol. 532

FIG. 4.ÈPDF conÐdence contours for the spatially Ñat time-variable " scalar Ðeld model, with potential V (/) P /~a, derived using the three SN Ia data sets. Panel a shows those derived from all the R98 SNe, while panel b is from the R98 SNe excluding the z \ 0.97 one, and panel c is for the P99 Ðt C data. Solid (dotted) lines in panels aÈc are the 1, 2, and 3 p contours from the SN data with (without) theH andt constraints (eqs. [3], [4]); the dotted lines here ^ 0 0 are the solid lines in Figs. 1aÈ1c. Panel d compares the 2 p limits from theH0 andt0 constraints used in conjunction with all R98 SNe (long-dashed lines), with R98 SNe excluding the z \ 0.97 one (short-dashed lines), and the P99 Ðt C SNe (dotted lines).

with a time-independent equation of state. However, the Figure 3 shows the e†ects of incorporating constraints two sets of conÐdence contours di†er signiÐcantly at larger based onH andt measurements (eqs. [3] and [4]). Panel [ 0 0 w. Since the scalar ÐeldwÕ eventually switches over to 1 a shows that adding theH0 constraint does not signiÐcantly (in the scalar ÐeldÈdominated epoch; RP), it is unclear what alter the contours derived from the SN Ia data alone. This is signiÐcance should be ascribed to this di†erence. We stress, expected, since the value ofH0 used here is very close to the however, that since the time-independent equation of state value that is indicated by the SN Ia data (see R98). \ ^ Ñuid model is an approximation to the time-variable " However, incorporating thet0 constraint, t0 12 1.3 scalar Ðeld model, constraints on model-parameter values Gyr at 1 p, does signiÐcantly shift the contours) (panel b). that are based on the Ñuid model approximation are only This is because the SN Ia data alone favor a higher t0, approximate (and possibly indicative). In passing, we note 14.2 ^ 1.7 Gyr (R98) or 14.5 ^ 1.0 (0.63/h) Gyr (P99). that Perlmutter et al. (1999b) useweff (eq. [2]) and not w to Figure 4 shows constraints on the time-variable " model, parameterize the Ñuid model constraints. Figure 2b shows from the three di†erent SN Ia data sets used in conjunction contours of constantweff as a function of a and)0 in the with theH0 andt0 measurements (eqs. [3] and [4]). The time-variable " scalar Ðeld model. In a large part of model main e†ect of incorporating theH0 andt0 constraints is to parameter spaceweff is a sensitive function of both a and )0 increase the favored values of)0 ; a weaker e†ect is the and hence is not the best parameter to use to describe the disfavoring of larger values of a. Even this extended set of time-variable " scalar Ðeld model. data does not tightly constrain model-parameter values. No. 1, 2000 SN Ia CONSTRAINTS ON COSMOLOGICAL ““ CONSTANT ÏÏ 115

FIG. 5.ÈPDF conÐdence contours for the time-independent " model, derived using the three SN Ia data sets. Panel a shows those derived from all the R98 SNe, while panel b is from the R98 SNe excluding the z \ 0.97 one and panel c is for the P99 Ðt C SNe. Solid (dotted) lines in panels aÈc are the 1, 2, and 3 ^ p contours from the SN data with (without) theH0 andt0 constraints (eqs. [3] and [4]). Panel d compares the 2 p limits from theH0 andt0 constraints used in conjunction with all R98 SNe (long-dashed lines), R98 SNe excluding the z \ 0.97 one (short-dashed lines), and the P99 Ðt C SNe (dotted lines). In all panels, models with parameter values in the upper left-hand corner region bounded by the diagonal dot-dashed curve do not have a big bang. The horizontal \ \ dot-dashed line demarcates models with a zero ", and the diagonal dot-dashed line running from)" 1 to)0 2 corresponds to spatially Ñat models. Figure 5 shows the corresponding constraints on the the incomplete understanding of a number of astrophysical constant-" model (from the SN Ia,H0, andt0 measure- e†ects and processes (evolution, intergalactic dust, etc.) ments). Again, the major e†ect of including theH0 and t0 means that these results are preliminary and not yet deÐni- data is to increase the favored values of)0. A weaker e†ect tive. is that it reduces the odds against reasonable open models The constraints on the time-variable " model derived (Go rski et al. 1998), but not by a large factor. here are based on the exact scalar Ðeld model equations of Figures 6 and 7 show the e†ects of using the nonin- motion, not on the widely used time-independent equation formative prior,1/)0, of equation (5). The major e†ect is a of state Ñuid approximation equations of motion. decrease in the favored values of)0. In the time-variable " case there is also a slight increase in the favored values of a We acknowledge the advice and assistance of R. Gott, (see Fig. 6), while in the constant-" case there is a mild B. Kirshner, L. Krauss, V. Periwal, S. Perlmutter, A. Riess, reduction in the odds against reasonable open models (see E. Sidky, M. Vogeley, and I. Wasserman, and are especially Fig. 7). If the PDF was narrower (i.e., if the error bars on the indebted to J. Peebles and T. Souradeep. We acknowledge data were smaller), changing from the Ñat to the nonin- helpful discussions with I. Waga, who has also computed formative prior would not result in as large a change in the the PDF conÐdence contours for the time-variable " scalar conÐdence contours. CONCLUSION Ðeld model, with results consistent with those found in this 4. paper. We thank the referee, I. Wasserman, for a prompt Recent SN Ia data do favor models with a constant or and detailed report that helped us improve the manuscript. time-variable " over an open model without a ". However, We acknowledge support from NSF CAREER grant AST this is not at a very high level of statistical signiÐcance. Also, 98-75031. FIG. 6.ÈPDF conÐdence contours (1, 2, and 3 p) derived from the P99 Ðt C SNe, for the spatially Ñat time-variable " scalar Ðeld model. Panel a(b) ignores (accounts for) theH0 andt0 constraints (eqs. [3] and [4]). The solid (dotted) lines use the noninformative1/)0 (Ñat) prior. The dotted lines in panel a(b) are the same as the solid lines in Fig. 1c (4c).

FIG. 7.ÈPDF conÐdence contours (1, 2, and 3 p) derived from the R98 SNe excluding the z \ 0.97 one, for the time-independent " model. Panel a(b) ignores (accounts for) theH0 andt0 constraints (eqs. [3] and ]4]). The solid (dotted) lines use the noninformative1/)0 (Ñat) prior. The dotted lines in panel a (b) are the same as the dotted (solid) lines in Fig. 5b. The dot-dashed lines are described in the caption of Fig. 5. In the noninformative prior cases the \ likelihood function is computed down to )0 0.01. SN Ia CONSTRAINTS ON COSMOLOGICAL ““ CONSTANT ÏÏ 117

REFERENCES Aguirre, A. N. 1999, ApJ, 512, L19 Ho Ñich, P., Wheeler, J. C., & Thielemann, F. K. 1998, ApJ, 495, 617 Amendola, L., Corasaniti, S., & Occhionero, F. 1999, preprint (astro-ph/ Hu, W., Eisenstein, D. J., Tegmark, M., & White, M. 1999, Phys. Rev. D, 9907222) 59, 023512 Bahcall, N. A., Ostriker, J. P., Perlmutter, S., & Steinhardt, P. J. 1999, Jain, D., Panchapakesan, N., Mahajan, S., & Bhatia, V. B. 1998, ApJ, Science, 284, 1481 (astro-ph/9807192) Baker, J. C., et al. 1999, MNRAS, 308, 1173 Jaynes, E. T. 1968, IEEE Trans. Sys. Sci. Cyb., SSC-4, 227 Banks, T., Dine, M., & Nelson, A. E. 1999, J. High Energy Phys., 9906, 014 Kamionkowski, M., & Kosowsky, A. 1999, Ann. Rev. Nucl. Part. Sci., in Barr, S. M. 1999, Phys. Lett. B, 454, 92 press Barrow, J. D., & Magueijo, J. 2000, ApJ, in press Kim, J. E. 1999, J. High Energy Phys., 9905, 022 Bartelmann, M., Huss, A., Colberg, J. M., Jenkins, A., & Pearce, F. R. 1998, Kolda, C. & Lyth, D. H. 1999, Phys. Lett. B, 458, 197 A&A, 330, 1 Kravtsov, A. V., & Klypin, A. A. 1999, ApJ, 520, 437 Bento, M. C., & Bertolami, O. 1999, Gen. Rel. Grav., 31, 1461 Liddle, A. R., & Scherrer, R. J. 1999, Phys. Rev. D, 59, 023509 Berger, J. O. 1985, Statistical Decision Theory and Bayesian Analysis (New Lineweaver, C. H., & Barbosa, D. 1998, ApJ, 496, 624 York: Springer) Lucchin, F., & Matarrese, S. 1985, Phys. Rev. D, 32, 1316 Biggs, A. D., Browne, I. W. A., Helbig, P., Koopmans, L. V. E., Wilkinson, Madore, B. F., et al. 1999, ApJ, 515, 29 P. N., & Perley, R. A. 1999, MNRAS, 304, 349 Masiero, A., Pietroni, M., & Rosati, F. 2000, Phys. Rev. D, 61, 023504 Bine truy, P. 1999, Phys. Rev. D, 60, 063502 Meneghetti, M., Bolzonella, M., Bartelmann, M., Moscardini, L., & BloomÐeld Torres, L. F., & Waga, I. 1996, MNRAS, 279, 712 Tormen, G. 2000, MNRAS, in press Bond, J. R., Ja†e, A. H., & Knox, L. 1998, Phys. Rev. D, 57, 2117 Olive, K. A., Steigman, G., & Walker, T. P. 2000, Phys. Rep., in press Brax, P., & Martin, J. 1999, Phys. Lett. B, 468, 45 Overduin, J. M. 1999, ApJ, 517, L1 Bucher, M., & Spergel, D. 1999, Phys. Rev. D, 60, 043505 O zer, M. 1999, ApJ, 520, 45 Caldwell, R. R., Dave, R., & Steinhardt, P. J. 1998, Phys. Rev. Lett., 80, O zer, M., & Taha, M. O. 1986, Phys. Lett. B, 171, 363 1582 Peebles, P. J. E. 1984, ApJ, 284, 439 Carroll, S. M. 1998, Phys. Rev. Lett., 81, 3067 ÈÈÈ. 1993, Principles of Physical Cosmology (Princeton: Princeton Chaboyer, B., Demarque, P., Kernan, P. J., & Krauss, L. M. 1998, ApJ, Univ. Press) 494, 96 Peebles, P. J. E., & Ratra, B. 1988, ApJ, 325, L17 (PR) Chiba, T., Sugiyama, N., & Nakamura, T. 1997, MNRAS, 289, L5 Peebles, P. J. E., & Vilenkin, A. 1999, Phys. Rev. D, 59, 063505 Choi, K. 1999, preprint (hep-ph/9902292) Periwal, V. 1999, preprint (astro-ph/9906253) Cole, S., Weinberg, D. H., Frenk, C. S., & Ratra, B. 1997, MNRAS, 289, 37 Perlmutter, S., et al. 1997, ApJ, 483, 565 Colley, W. N., Gott, J. R., Weinberg, D. H., Park, C., & Berlind, A. A. 2000, ÈÈÈ. 1999a, ApJ, 517, 565 (P99) ApJ, 529, 795 Perlmutter, S., Turner, M. S., & White, M. 1999b, Phys. Rev. Lett., 83, 670 Cooray, A. R. 1999, A&A, 341, 653 Perrotta, F., & Baccigalupi, C. 1999, Phys. Rev. D, 59, 123508 Copeland, E. J., Liddle, A. R., & Wands, D. 1998, Phys. Rev. D, 57, 4686 Ratra, B. 1991, Phys. Rev. D, 43, 3802 Donahue, M., & Voit, G. M. 1999, ApJ, 523, L137 ÈÈÈ. 1992, Phys. Rev. D, 45, 1913 Doroshkevich, A. G.,Mu ller, V., Retzla†, J., & Turchaninov, V. 1999, Ratra, B., Ganga, K., Stompor, R., Sugiyama, N., de Bernardis, P., & MNRAS, 306, 575 Go rski, K. M. 1999a, ApJ, 510, 11 Drell, P. S., Loredo, T. J., & Wasserman, I. 2000, ApJ, in press Ratra, B., & Peebles, P. J. E. 1988, Phys. Rev. D, 37, 3406 (RP) Efstathiou, G. 1999, MNRAS, 310, 842 ÈÈÈ. 1994, ApJ, 432, L5 Efstathiou, G., Sutherland, W. J., & Maddox, S. J. 1990, Nature, 348, 705 ÈÈÈ. 1995, Phys. Rev. D, 52, 1837 Falco, E. E., Kochanek, C. S., & Munoz, J. A. 1998, ApJ, 494, 47 Ratra, B., & Quillen, A. 1992, MNRAS, 259, 738 (RQ) Ferreira, P. G., & Joyce, M. 1998, Phys. Rev. D, 58, 023503 Ratra, B., Stompor, R., Ganga, K., Rocha, G., Sugiyama, N., & Go rski, Fischler, W., Ratra, B., & Susskind, L. 1985, Nucl. Phys. B, 259, 730 K. M. 1999b, ApJ, 517, 549 Freese, K., Adams, F. C., Frieman, J. A., & Mottola, E. 1987, Nucl. Phys. B, Reid, I. N. 1997, AJ, 114, 161 287, 797 Riess, A. G., et al. 1998, AJ, 116, 1009 (R98) Freudling, W., et al. 1999, ApJ, 523, 1 Riess, A. G., Filippenko, A. V., Li, W., & Schmidt, B. P. 1999, AJ, 118, 2668 Frieman, J. A., Hill, C. T., Stebbins, A., & Waga, I. 1995, Phys. Rev. Lett., Rocha, G. 1999, in Dark Matter in Astrophysics and Particle Physics 1998, 75, 2077 ed. H. V. Klapdor-Kleingrothaus & L. Baudis (Bristol: IOP), 238 Frieman, J. A., & Waga, I. 1998, Phys. Rev. D, 57, 4642 Rocha, G., Stompor, R., Ganga, K., Ratra, B., Platt, S. R., Sugiyama, N., & Ganga, K., Ratra, B., Gundersen, J. O., & Sugiyama, N. 1997, ApJ, 484, 7 Go rski, K. M. 1999, ApJ, 525, 1 Ganga, K., Ratra, B., Lim, M. A., Sugiyama, N., & Tanaka, S. T. 1998, Sahni, V., & Starobinsky, A. 1999, Int. J. Mod. Phys. D, in press ApJS, 114, 165 Santiago, D. I., & Silbergleit, A. S. 1998, Phys. Lett. A, submitted Ganga, K., Ratra, B., & Sugiyama, N. 1996, ApJ, 461, L61 Stompor, R.,Go rski, K. M., & Banday, A. J. 1995, MNRAS, 277, 1225 Garc• a-Berro, E., Gaztanaga, E., Isern, J., Benvenuto, O., & Althaus, L. Suntze†, N. B., et al. 1999, AJ, 117, 1175 1999, preprint (astro-ph/9907440) Turner, M. S., & White, M. 1997, Phys. Rev. D, 56, 4439 Garnavich, P. M., et al. 1998, ApJ, 509, 74 Umeda, H., Nomoto, K., Kobayashi, C., Hachisu, I., & Kato, M. 1999, Garriga, J., Livio, M., & Vilenkin, A. 2000, Phys. Rev. D, 61, 023503 ApJ, 522, L43 Giovannini, M. 1999, Phys. Rev. D, 60, 123511 Uzan, J.-P. 1999, Phys. Rev. D, 59, 123510 Go rski, K. M., Ratra, B., Stompor, R., Sugiyama, N., & Banday, A. J. 1998, Waga, I., & Miceli, A. P. M. R. 1999, Phys. Rev. D, 59, 103507 ApJS, 114, 1 Wang, L., Caldwell, R. R., Ostriker, J. P., & Steinhardt, P. J. 2000, ApJ, in Go rski, K. M., Ratra, B., Sugiyama, N., & Banday, A. J. 1995, ApJ, 444, press L65 Wang, Y. 2000, ApJ, in press Gott, J. R. 1982, Nature, 295, 304 Wetterich, C. 1995, A&A, 301, 321 ÈÈÈ. 1997, in Critical Dialogues in Cosmology, ed. N. Turok Yamamoto, K., & Bunn, E. F. 1996, ApJ, 464, 8 (Singapore: World ScientiÐc), 519 Zlatev, I., Wang, L., & Steinhardt, P. J. 1999, Phys. Rev. Lett., 82, 896 Gratton, R. G., Fusi Pecci, F., Carretta, E., Clementini, G., Corsi, C. E., & Lattanzi, M. 1997, ApJ, 491, 749