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Supernova Ia Constraints on a Time-Variable Cosmological THE ASTROPHYSICAL JOURNAL, 532:109È117, 2000 March 20 ( 2000. The American Astronomical Society. All rights reserved. Printed in U.S.A. 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 uni- verse 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.
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