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66. The mean equatorial radius was derived from the tics for providing the engineering foundation that such a young age by such a wide array of harmonic model (65) based on a 1° sampling of an enabled this analysis. We also thank G. Elman, P. recent independent measurements. equatorial profile. This value is 200 m larger than was Jester, and J. Schott for assistance in altimetry pro- estimated from earlier data (5) but is within the error cessing, D. Rowlands and S. Fricke for help with orbit estimate of the earlier value. The uncertainty corre- determination, S. Zhong for assistance with the Hel- Method sponds to the standard error of the mean of the 360 las relaxation calculation, and G. McGill for a con- ⍀ Any measurement of a function of h, m, and equatorial samples. structive review. The MOLA investigation is support- ⍀ 67. We acknowledge the MOLA instrument team and the ed by the NASA Mars Global Surveyor Project. ⌳ can be included in a joint likelihood MGS spacecraft and operation teams at the Jet Pro- N pulsion Laboratory and Lockheed-Martin Astronau- 21 April 1999; accepted 10 May 1999 ͑ ⍀ ⍀ ͒ ϭ ͹ L h, m, ⌳ Li (1) iϭ1 which I take as the product of seven of the most recent independent cosmological con- A Younger Age for the straints (Table 1 and Fig. 3). For example,

one of the Li in Eq. 1 represents the con- Charles H. Lineweaver straints on h. Recent measurements can be summarized as h៮ ϭ 0.68 Ϯ 0.10 (16). I The age of the universe in the model can be calculated from three represent these measurements in Eq. 1 by the ⍀ likelihood parameters: Hubble’s constant, h; the mass density of the universe, m; and the ⍀ ៮ 2 , ⌳. Recent observations of the cosmic h Ϫ h ͑ ͒ ϭ ͫϪ ͩ ͪ ͬ background and six other cosmological measurements reduce the uncertainty LHubble h exp 0.5 Ϯ 0.10 in these three parameters, yielding an age for the universe of 13.4 1.6 billion (2) , which is a younger than other recent age estimates. A different standard Big Bang model, which includes cold with a Another Li in Eq. 1 comes from measure- cosmological constant, provides a consistent and absolutely time-calibrated ments of the fraction of normal baryonic evolutionary sequence for the universe. matter in clusters of (14) and esti- mates of the density of normal baryonic mat- ⍀ 2 ϭ Ϯ In the Big Bang model, the age of the uni- ments with SNe and other data, I (9) have ter in the universe [ bh 0.015 0.005 ⍀ ϭ Ϯ verse, to, is a function of three parameters: h, reported ⌳ 0.62 0.16 [see (10–12) for (15, 18)]. When combined, these measure- ⍀ ⍀ ⍀ Þ ⍀ 2/3 ϭ Ϯ m, and ⌳ (1). The dimensionless Hubble similar results]. If ⌳ 0, then estimates of the ments yield mh 0.19 0.12 (19), constant, h, tells us how fast the universe is age of the universe in Big Bang models must which contributes to the likelihood through ⍀ expanding. The density of matter in the uni- include ⌳. Thus, one must use the most gen- ͑ ⍀ ͒ Lbaryons h, m verse, ⍀ , slows the expansion, and the cos- eral form: t ϭ f(⍀ , ⍀ )/h (13). m o m ⌳ ⍀ 2/3 Ϫ ⍀ 2/3 2 ⍀ mh mh mological constant, ⌳, speeds up the expan- Here, I have combined recent independent ϭ expͫϪ0.5ͩ ͪ ͬ sion (Fig. 1). measurements of CMB anisotropies (9), type Ia 0.12 Until recently, large uncertainties in the SNe (4, 5), cluster mass-to-light ratios (6), clus- (3) ⍀ ⍀ ⍀ ⍀ measurements of h, m, and ⌳ made efforts ter abundance evolution (7), cluster baryonic The ( m, ⌳)–dependencies of the remain- ⍀ ⍀ to determine to (h, m, ⌳) unreliable. The- fractions (14), deuterium-to-hydrogen ratios in ing five constraints are plotted in Fig. 3 (20). oretical preferences were, and still are, often quasar spectra (15), double-lobed radio sources The 68% confidence level regions derived used to remedy these observational uncertain- (8), and the Hubble constant (16) to determine from CMB and SNe (Fig. 3, A and B) are ⍀ ϭ ties. One assumed the standard model ( m the age of the universe. The big picture from the nearly orthogonal, and the region of overlap 1, ⍀⌳ ϭ 0), dating the age of the universe to analysis done here is as follows (Figs. 1 and 2): is relatively small. Similar complementarity ϭ ϳ to 6.52/h billion years old (Ga). However, The Big Bang occurred at 13.4 Ga. About 1.2 exists between the CMB and the other data for large or even moderate h estimates billion years (Gy) later, the halo of our sets (Fig. 3, C through E). The combination

(Ռ0.65), these simplifying assumptions re- (and presumably the halo of other galaxies) of them all (Fig. 3F) yields ⍀⌳ ϭ 0.65 Ϯ ⍀ ϭ Ϯ sulted in an age crisis in which the universe formed. About 3.5 Gy later, the disk of our 0.13 and m 0.23 0.08 (21). Ϸ Ͻ was younger than our Galaxy (to 10 Ga Galaxy (and presumably the disks of other This complementarity is even more im- Ϸ tGal 12 Ga). These assumptions also result- spiral galaxies) formed. This picture agrees portant (but more difficult to visualize) in ⍀ ed in a baryon crisis in which estimates of the with what we know about galaxy forma- three-dimensional parameter space, (h, m, amount of normal (baryonic) matter in the tion. Even the recent indications of the ⍀⌳). Although the CMB alone cannot tightly universe were in conflict (2, 3). existence of old galaxies at high constrain any of these parameters, it does ⍀ Ͻ Evidence in favor of m 1 has become (17) fit into the time framework determined have a strong preference in the three-dimen- ⍀ ⍀ ⍀ more compelling (4–8), but ⌳ is still often here. In this sense, the result is not surpris- sional space (h, m, ⌳). In Eq. 1, I used ⍀ ⍀ assumed to be zero, not because it is measured ing. What is new is the support given to LCMB(h, m, ⌳), which is a generalization to be so, but because models are simpler with- out it. Recent evidence from supernovae (SNe) (4, 5) indicates that ⍀ Ͼ 0. These SNe data Table 1. Parameter estimates from non-CMB measurements. I refer to these as constraints. I use the error ⌳ bars cited here as 1␴ errors in the likelihood analysis. The first four constraints are plotted in Fig. 3, B and other data exclude the standard Einstein- through E. ⍀ ϭ ⍀ ϭ deSitter model ( m 1, ⌳ 0). The cosmic microwave background (CMB), on the other Method Reference Estimate ⍀ ⍀ ϭ hand, excludes models with low m and ⌳ ⍀ ⍀ ⍀⌳ϭ0 ϭϪ Ϯ ⍀flat ϭ Ϯ 0(3). With both high and low m excluded, ⌳ SNe (35) m 0.28 0.16, m 0.27 0.14 ⍀⌳ϭ0 ϭ Ϯ cannot be zero. Combining CMB measure- Cluster mass-to-light (6) m 0.19 0.14 ⍀⌳ϭ0 ϭ ϩ0.28 ⍀flat ϭ ϩ0.25 Cluster abundance evolution (7) m 0.17Ϫ0.10 , m 0.22Ϫ0.10 ⍀⌳ϭ0 ϭϪ ϩ0.70 ⍀flat ϭ ϩ0.50 Double radio sources (8) m 0.25Ϫ0.50 , m 0.1Ϫ0.20 ⍀ 2/3 ϭ Ϯ School of Physics, University of New South Wales, Baryons (19) mh 0.19 0.12 Sydney NSW 2052, Australia. E-mail: charley@bat. Hubble (16) h ϭ 0.68 Ϯ 0.10 phys.unsw.edu.au

www.sciencemag.org SCIENCE VOL 284 28 MAY 1999 1503 R ESEARCH A RTICLES ⍀ ⍀ ϭ Ϯ of LCMB( m, ⌳) (Fig. 3A) (22). To convert h 0.68 0.10 but does not depend strongly age, independent of the SNe and CMB data, ⍀ Ϫ Ϫ ϭ ϩ3.4 the three-dimensional likelihood L(h, m, on the central value assumed for Hubble’s con- is to(all CMB SNe) 12.6Ϫ2.0 Ga, ⍀⌳) of Eq. 1 into an estimate of the age of the stant (as long as this central value is in the most which is somewhat lower than the main result universe and into a more easily visualized accepted range, 0.64 Յ h Յ 0.72) or on the but within the error bars.

two-dimensional likelihood, L(h, to), I com- uncertainty of h (unless this uncertainty is taken puted the dynamic age corresponding to each to be very small). Assuming an uncertainty of The Oldest Objects in Our Galaxy ⍀ ϭ point in the three-dimensional space (h, m, 0.10, age estimates from using h 0.64, 0.68, The universe cannot be younger than the oldest ⍀ ⌳). For a given h and to, I then set L(h, to) and 0.72 are 13.5, 13.4, and 13.3 Ga, respec- objects in it. Thus, estimates of the age of the ⍀ ⍀ equal to the maximum value of L(h, m, ⌳) tively (Fig. 2). Using a larger uncertainty of oldest objects in our Galaxy are lower limits to 0.15 with the same h values does not substan- the age of the universe (Table 2 and Fig. 2). A L͑h, t ͒ ϭ max͓L͑h, ⍀ , ⍀⌳͒ ͑ ⍀ ⍀ ͒Ϸ ͔ o m t h, m, ⌳ to tially change the results, which are 13.4, 13.3, standard but simplified scenario for the origin (4) and 13.2 Ga, respectively. For both groups, the of our Galaxy has a halo of globular clusters ϰ This has the advantage of explicitly display- age difference is only 0.2 Gy. If to 1/h were forming first, followed by the formation of the ing the h dependence of the to result. The adhered to, this age difference would be 1.6 Gy. Galactic disk. The most recent measurements of joint likelihood L(h, to) of Eq. 4 yields an age Outside the most accepted range, the h depen- the age of the oldest objects in the Galactic disk ϭ Ϯ ϰ ϭ Ϯ for the universe of to 13.4 1.6 Ga (Fig. dence becomes stronger and approaches to give tdisk 8.7 0.4 Ga (Table 2). The most 4). This result is a billion years younger than 1/h (23). recent measurements of the age of the oldest ϭ other recent age estimates. To show how each constraint contributes objects in the halo of our Galaxy give tGal Ϯ What one uses for LHubble(h)inEq.1is to the result, I convolved each constraint 12.2 0.5 Ga (Table 2). The individual mea- particularly important because, in general, we separately with Eq. 2 (Fig. 5). The result does surements are in good agreement with these expect the higher h values to yield younger not depend strongly on any one of the con- averages. There are no large outliers. In contrast Ϫ ⍀ ⍀ ages. Table 2 contains results from a variety of straints (see “all x” results in Table 2). For to the to(h, m, ⌳) estimates obtained above, h estimates, assuming various central values example, the age, independent of the SNe all of these age estimates are direct in the sense Ϫ ϭ ϩ1.7 and various uncertainties around these values. data, is to(all SNe) 13.3Ϫ1.8 Ga, which that they have no dependence on a Big Bang ϭ Ϯ The main result of to 13.4 1.6 Ga has used differs negligibly from the main result. The model.

Fig. 1 (left). The size of the universe, in units of its current size, as a function of time. The age of the five models can be read from the x axis as the time between NOW and the intersection of the model with the x ϭ Ϯ ϭ axis. The main result of this paper, to 13.4 1.6 Ga, is labeled “to” and three similar points near 13.4 Ga result from h 0.64, 0.68, and 0.72 is shaded gray on the x axis. Measurements of the age of the halo of our and indicate that the result is not strongly dependent on h when a ϭ Ϯ Ϯ Galaxy yield tGal 12.2 0.5 Ga, whereas measurements of the age of reasonable h uncertainty of 0.10 is used. Among the four, the highest the disk of our Galaxy yield t ϭ 8.7 Ϯ 0.4 Ga ( Table 2). These age value at 14.6 Ga comes from assuming h ϭ 0.64 Ϯ 0.02. All the disk ⍀ ⍀ ϭ ranges are also labeled and shaded gray. The ( m, ⌳) (0.3, 0.7) model estimates in the top section of Table 2 are plotted here. As in Fig. 1, fits the constraints of Table 1 better than the other models shown. Over averages of the ages of the Galactic halo and Galactic disk are shaded the past few billion years and on into the future, the rate of expansion gray. The absence of any single age estimate more than ϳ2␴ from the of this model increases (R¨ Ͼ 0). This acceleration means that we are in average adds plausibility to the possibly overdemocratic procedure of ⍀ Ͼ a period of slow inflation. Other consequences of a ⌳-dominated computing the variance-weighted averages. The result that to tGal is universe are discussed in (50). On the x axis, h ϭ 0.68 has been logically inevitable, but the standard Einstein-deSitter model does not assumed. For other values of h, multiply the x axis ages by 0.68/h. satisfy this requirement unless h Ͻ 0.55. The reference for each are indicated on the right. Fig. 2 (right). Age estimates of measurement is given under the x axis. The age of the is accurately the universe and of the oldest objects in our Galaxy. The four estimates known and is included for reference. Error bars indicate the reported 1␴ of the age of the universe from this work are indicated in Table 2. The limits.

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m ⍀ ⍀ Fig. 3 (left). The regions of the ( m, ⌳) plane preferred by various con- straints. (A) Cosmic microwave background, (B) SNe, (C) cluster mass-to-light ratios, (D) cluster abundance evolution, (E) double radio lobes, and (F) all combined. The power of combining CMB constraints with each of the other constraints (Table 1) is also shown. The elongated areas (from upper left to lower right) in (A) are the approximate 1␴,2␴, and 3␴ confidence levels of the likelihood from CMB data, LCMB (9). (A) also shows the important h depen- dence of L . The contours within the dark shaded region are of h values that CMB ⍀ ⍀ ϭ maximize LCMB for a given ( m, ⌳) pair (h 0.70 and 0.90). This correlation between preferred h and preferred (⍀ , ⍀⌳) helps L (h, ⍀ , ⍀⌳) constrain m ␴ CMB m ␴ to. In (B) through (E), thin contours enclose the 1 (shaded) and 2 confidence regions from separate constraints, and thick contours indicate the 1␴,2␴, and ␴ 3 regions of the combination of LCMB with these same constraints. (F) shows the region preferred by the combination of the separate constraints shown in (B) through (E) (thin contours) as well as the combination of (A) through (E) ⍀ ϭ Ϯ ⍀ ϭ Ϯ (thick contours). The best fit values are ⌳ 0.65 0.13 and m 0.23 0.08. In (A), the thin iso-to contours (labeled “10” through “14”) indicate the age (billion years ago) when h ϭ 0.68 is assumed. For reference, the 13- and 14-Ga contours are in all panels. To give an idea of the sensitivity of the h dependence of these contours, the two additional dashed contours in (A) show the 13-Ga contours for h ϭ 0.58 and 0.78 (the 1␴ limits of the principle Ϸ h estimate used in this paper). In (F), it appears that the best fit has to 14.5 Ga, but all constraints shown here are independent of information about h; they do not include the h dependence of LCMB, Lbaryons,orLHubble (Table 1). Fig. 4 (top right). This plot shows the region of the h Ϫ t plane ϭ o Ϯ preferred by the combination of all seven constraints. The result, to 13.4 1.6 Ga, is the main result of this paper. The thick contours around the best fit ϭ (indicated by a ) are at likelihood levels defined by L/Lmax 0.607 and 0.135, which approximate 68 and 95% confidence levels, respectively. These contours can be projected onto the t axis to yield the age result. This age result is robust to variations in the Hubble constraint as o Յ ⍀ Յ Յ indicated in Table 2. The areas marked “Excluded” (here and in Fig. 5) result from the range of parameters considered: 0.1 m 1.0 and 0 ⍀⌳ Յ 0.9 with ⍀ ϩ⍀⌳ Յ 1. Thus, the upper (high t ) boundary is defined by (⍀ , ⍀⌳) ϭ (0.1, 0.9), and the lower boundary is the standard m ⍀ ⍀ ϭ o m Einstein-deSitter model defined by ( m, ⌳) (1, 0). Both of these boundary models are plotted in Fig. 1. The estimates from Table 2 of the age of our Galactic halo (tGal) and the age of the (tdisk) are shaded grey. The universe is about 1 billion years older than our Galactic halo. The combined constraints also yield a best fit value of the Hubble constant which can be read off of the x axis (h ϭ 0.73 Ϯ 0.09, a slightly higher and tighter estimate than the input h ϭ 0.68 Ϯ 0.10). Fig. 5 (bottom right). The purpose of this figure is to show how Fig. 4 is built up from the seven independent constraints used in the analysis. All six panels are analogous to Fig. 4 but contain only the Hubble constraint [h ϭ 0.68 Ϯ 0.10 (Eq. 2)] convolved with a single constraint: (A) cosmic microwave background, (B) SNe, (C) cluster mass-to-light ratios, (D) cluster abundance evolution, (E) double radio lobes, and (F) baryons (Table 1). The relative position of the best fit (indicated by a star) and the 13.4-Ga line indicates how each constraint contributes to the result.

www.sciencemag.org SCIENCE VOL 284 28 MAY 1999 1505 R ESEARCH A RTICLES How old was the universe when our Galaxy between the age reported here and the estimate fewer disks in the halolike progenitors of ϩ⌬ ϭ ⌬ ϭ formed? If we write this as tGal t to, then of the age of our Galaxy (Table 2). Thus, t spiral galaxies in this redshift range. Studies ⌬ Ϫ ϭ Ϫ ϭ Ϯ what is the amount of time ( t) between the to tGal 13.4 12.2 1.2 1.8 Gy. of galaxy types in the formation of our Galaxy and the formation of The age measurements in Table 2 also indicate that this may be the case (26). the universe? If we had an estimate of ⌬t, then indicate that there is a 3.5-Gy period between The requirement that the universe be older Ϫ Ͼ we would have an independent estimate of to to halo and disk formation (tGal tdisk). If our than our Galaxy (to tGal) is a consistency ϭ Ϯ compare to to 13.4 1.6 Ga, obtained above. Milky Way is typical, then this may be true of test of the Big Bang model. The best fit However, we have very poor constraints on ⌬t. other spiral galaxies. With the best fit values model obtained here passes this test. There is The simple but plausible estimate ⌬t Ϸ 1Gyis obtained here for the three parameters, (h, no age crisis. This is true even if the high ϳ ⍀ ⍀ ϭ Ϯ Ϯ ϳ Ϸ often invoked, but estimates range from 0.1 m, ⌳) (0.72 0.09, 0.23 0.08, values of h ( 0.80) are correct. Only at h ϳ Ϯ Ϸ to 5 Gy and may be even larger (24, 25). This 0.65 0.13), the ages tdisk and tGal can be 0.85 is to tGal. This consistency provides uncertainty in ⌬t undermines the ability of es- converted into the redshifts at which the disk further support for the Big Bang model, ϭ ϩ1.5 ϭ ⍀ ϭ ⍀ ϭ timates of the age of the oldest objects in our and halo formed: zdisk 1.3Ϫ0.5 and zGal which the standard model ( m 1, ⌳ 0) ϩϱ Ͻ Galaxy to tell us the age of the universe. With- 6.0Ϫ4.3. Thus, a diskless epoch should be is unable to match unless h 0.55. ⌬ out t, we cannot infer to from tGal. The best centered at a redshift between zdisk and zGal ⌬ գ գ Comparison with Previous Work estimate of t may come from the difference (1.3 zdiskless 6.0). We would expect The goal of this paper is to determine the ⍀ ⍀ absolute age of the universe to(h, m, ⌳). Table 2. Age estimates of our Galaxy and universe (36). “Technique” refers to the method used to make Knowledge of h alone cannot be used to deter- the age estimate. OC, open clusters; WD, dwarfs; LF, luminosity function; avg, average; GC, globular clusters; M/L, mass-to-light ratio; and cl evol, cluster abundance evolution. The averages are inverse mine to with much accuracy. For example, the variance-weighted averages of the individual measurements. The sun is not included in the disk average. estimate h ϭ 0.68 Ϯ 0.10 corresponds to 8 Ͻ Ͻ “Isotopes” refers to the use of relative isotopic abundances of long-lived species as indicated by to 22 Ga (Fig. 4). Similarly, knowledge of absorption lines in spectra of old disk . The “stellar ages” technique uses fitting and ⍀ ⍀ ⍀ ⍀ ( m, ⌳) yields Hoto( m, ⌳), not to (Ho is the the new Hipparcos subdwarf calibration. “All” means that all six constraints in Table 1 and the CMB Ϫ usual Hubble constant). When one inserts a constraints were used in Eq. 1. “All x” means that all seven constraints except constraint x were used preferred value of h into a H t result, one is not in Eq. 1. Figures 3 and 5 and the all Ϫ x results indicate a high level of agreement between constraints o o and the lack of dependence on any single constraint. Thus, there is a broad consistency between the ages taking into consideration the correlations be- ⍀ preferred by the CMB and the six other independent constraints. Figure 2 presents all of the disk and halo tween preferred h values and preferred ( m, age estimates. ⍀⌳) values that are inherent, for example, in ⍀ ⍀ ⍀ LCMB(h, m, ⌳) and Lbaryons(h, m). The Technique Reference h assumptions Age (Ga) Object preferred values of h in these likelihoods de- ⍀ ⍀ Ϯ pend on m and ⌳. Perlmutter et al.(4) used Isotopes (37) None 4.53 0.04 Sun ⍀ ⍀ SNe measurements to constrain ( m, ⌳) and obtained values for H t . To obtain t , they did Stellar ages (38) None 8.0 Ϯ 0.5 Disk OC o o o WD LF (39) None 8.0 Ϯ 1.5 Disk WD the analysis with h set equal to the value pre- Stellar ages (40) None 9.0 Ϯ 1 Disk OC ferred by their SNe data, h ϭ 0.63. Their result ϩ0.9 ϭ Ϯ WD LF (25) None 9.7Ϫ0.8 Disk WD is to 14.5 1.0(0.63/h) Ga. When a flat ϩ1.0 Stellar ages (41) None 12.0Ϫ Disk OC flat ϭ 2.0 universe is assumed, they obtain to Ϯ ϩ None 8.7 0.4 tdisk (avg) 1.4 ϭ 14.9Ϫ1.1(0.63/h) Ga. Riess et al.(5) found h Ϯ Ϯ 0.65 0.02 from their SNe data. Marginalizing Stellar ages (42) None 11.5 1.3 Halo GC ⍀ ⍀ ϩ1.1 over this Hubble value and over ⌳ and m, Stellar ages (43) None 11.8Ϫ Halo GC 1.3 ϭ Ϯ Stellar ages (44) None 12 Ϯ 1 Halo GC they report to 14.2 1.7 Ga. When a flat flat ϭ Stellar ages (45) None 12 Ϯ 1 Halo GC universe is assumed, their results yield to Stellar ages (46) None 12.5 Ϯ 1.5 Halo GC 15.2 Ϯ 1.7 Ga. The Perlmutter et al.(4) and Isotopes (47) None 13.0 Ϯ 5 Halo stars Riess et al.(5) results are in good agreement. Stellar ages (48) None 13.5 Ϯ 2 Halo GC ϭ Ϯ ϭ ϩ2.3 When I assume h 0.64 0.02, I get to Stellar ages (49) None 14.0Ϫ Halo GC ϩ 1.6 14.6 1.6 Ga. This result is plotted in Fig. 2 to None 12.2 Ϯ 0.5 t (avg) Ϫ1.1 Gal illustrate the important influence on the result of SNe (4) 0.63 Ϯ 0.0 14.5 Ϯ 1.0 Universe using a small h uncertainty. Efstathiou et al. Ϯ ϩ1.4 (12), on the basis of a combination of CMB and SNe (flat) (4) 0.63 0.0 14.9Ϫ1.1* Universe SNe (5) 0.65 Ϯ 0.02 14.2 Ϯ 1.7 Universe Perlmutter et al.(4) SNe data, have estimated SNe (flat) (5) 0.65 Ϯ 0.02 15.2 Ϯ 1.7* Universe ϭ Ϫ1 ϭ Ϯ ϩ to 14.6(h/0.65) Ga. I used h 0.65 0.0 Ϯ 2.3 ϩ1.2 All This work 0.60 0.10 15.5Ϫ2.8 Universe with this data combination to get t ϭ 14.5 Ϯ ϩ3.5 o Ϫ1.0 All This work 0.64 0.10 13.5Ϫ2.2* Universe ϭ Ϯ Ϯ ϩ1.6 Ga. However, when I used h 0.65 0.10, the All This work 0.68 0.10 13.4Ϫ * Universe ϩ ϩ1.6 ϭ 3.2 Ϯ 1.2 result is 0.7 Gy lower (to 13.8Ϫ1.4 Ga). To All This work 0.72 0.10 13.3Ϫ1.9* Universe Ϯ ϩ1.9 obtain the main result, I used uncertainties large All This work 0.76 0.10 12.3Ϫ1.6 Universe Ϯ ϩ1.9 All This work 0.80 0.10 11.9Ϫ1.6 Universe enough to reflect our knowledge of h,onthe Ϯ ϩ1.6 All This work 0.64 0.02 14.6Ϫ1.1* Universe basis of many sources. The use of a larger h Ϫ Ϯ ϩ3.0 All CMB This work 0.68 0.10 14.0Ϫ2.2 Universe uncertainty contributes to the substantially Ϫ Ϯ ϩ1.7 All SNe This work 0.68 0.10 13.3Ϫ1.8 Universe Ϫ Ϯ ϩ1.9 younger ages found here (23). All M/L This work 0.68 0.10 13.3Ϫ1.7 Universe Ϫ Ϯ ϩ1.7 A potential problem with the SNe ages is All cl evol This work 0.68 0.10 13.3Ϫ1.4 Universe Ϫ Ϯ ϩ1.7 the high region, (⍀ , ⍀ ) Ϸ (0.8, 1.5), which All radio This work 0.68 0.10 13.3Ϫ1.5 Universe m ⌳ Ϫ Ϯ ϩ2.6 All baryons This work 0.68 0.10 13.4Ϫ1.5 Universe dominates the SNe fit. This region is strongly All Ϫ Hubble This work None Ͻ14.2 Universe disfavored by the six other constraints consid- Ϫ Ϫ Ϯ ϩ3.4 All CMB SNe This work 0.68 0.10 12.6Ϫ2.0 Universe ⍀ ⍀ ered here (see Fig. 3). These high ( m, ⌳) flat *Also plotted in Fig. 2. values allow lower ages than the to SNe re-

1506 28 MAY 1999 VOL 284 SCIENCE www.sciencemag.org R ESEARCH A RTICLES ϰ sults because the slope of the iso-to con- 12. G. Efstathiou, S. L. Bridle, A. N. Lasenby, M. P. Hobson, and 13.5 Ga, respectively (Fig. 2). If to 1/h were tours (Fig. 3B) is larger than the slope of R. S. Ellis, Mon. Not. R. Astron. Soc. 303, 47 (1999). adhered to, there would be no difference. flat 13. J. E. Felten and R. Isaacman, Rev. Mod. Phys. 58, 689 24. P. B. Stetson, D. A. VandenBerg, M. Bolte, Publ. As- the SNe contours. The to results are not as (1986); S. M. Carroll, W. H. Press, E. L. Turner, Annu. tron. Soc. Pac. 108, 560 (1996). subject to this problem and are the results Rev. Astron. Astrophys. 30, 499 (1992), Eq. 17. There 25. T. D. Oswalt, J. A. Smith, M. A. Wood, P. Hintzen, most analogous to the result reported here, may be some ambiguity about what the “age of the Nature 382, 692 (1996); T. D. Oswalt, personal com- flat universe” really means. I have used the Friedman munication. Together, these references provided the despite the fact that the SNe to results are equation derived from to extract age that is referred to as “Oswalt 98” in Fig. 2. less consistent with the result reported here. the age of the universe. This age is the time that has 26. S. P. Driver et al., Astrophys. J. 496, L93 (1998). But There are several independent cosmologi- elapsed since the early moments of the universe there is no consensus on this issue. See F. R. Marleau when the classical equations of general relativity and L. Simard, ibid. 507, 585 (1998). cal measurements that have not been in- became valid. In eternal inflation models [see, for 27. E. E. Falco, C. S. Kochanek, J. A. Munoz, ibid. 494,47 cluded in this analysis either because a example, A. Linde, Particle Physics and Inflationary (1998). consensus has not yet been reached [grav- (Harwood Academic, Chur, Switzerland, 28. Y. N. Cheng and L. M. Krauss, in preparation (available itational lensing limits (27–30)] or because 1990)] or other multiple universe scenarios (“multi- at http://xxx.lanl.gov/abs/astro-ph/9810393). verse”), the age of the universe is more complicated. 29. A. R. Cooray, in preparation (available at http://xxx. the analysis of the measurements has not If we live in a , then the age computed here lanl.gov/abs/astro-ph/9811448). been done in a way that is sufficiently free refers only to the age of our bubblelike part of it. 30. R. Quast and P. Helbig, in preparation (available at of conditioning on certain parameters [local 14. A. E. Evrard, Mon. Not. R. Astron. Soc. 292, 289 http://xxx.lanl.gov/abs/astro-ph/9904174). (1997). 31. I. Zehavi, paper presented at the MPA/ESO Cosmol- velocity field limits (31)]. Doubts about 15. S. Burles and D. Tytler, Astrophys. J. 499, 699 (1998). ogy Conference, Evolution of Large-Scale Structure: some of the observations used here are 16. B. F. Madore et al. (available at http://xxx.lanl.gov/ From Recombination to Garching, 2 to 7 August ϭ Ϯ 1998 (available at http://xxx.lanl.gov/abs/astro-ph/ discussed in (32). There has been specula- abs/astro-ph/9812157) report h 0.72( 0.05)r Ϯ 9810246). ⍀ ( 0.12)s from an analysis that combines recent Ho tion recently that the evidence for ⌳ is measurements. The errors are random and system- 32. A. Dekel, D. Burstein, S. D. M. White, in Critical really evidence for some form of stranger atic, respectively. F. Hoyle, G. Burbidge, and J. V. Dialogues in Cosmology N. Turok, Ed. (World Scien- that we have been incorrectly Narlikar [Mon. Not. R. Astron. Soc. 286, 173, 1997] tific, River Edge, NJ, 1998), pp. 175–191. ⍀ reviewed the observational literature and concluded 33. P. M. Garnavich et al., Astrophys. J. 509, 74 (1998). interpreting as ⌳. Several workers have ϭ ϩ0.10 ϭ Ϯ ϭ h 0.58Ϫ0.05. Both h 0.65 0.10 and h 34. S. Perlmutter, M. S. Turner, M. White, available at tested this idea. The evidence so far indi- 0.70 Ϯ 0.10 are often cited as standard estimates http://xxx.lanl.gov/abs/astro-ph/9901052. cates that the cosmological constant inter- [see, for example, (12, 18)]. I adopt h ϭ 0.68 Ϯ 0.10 35. I use SNe constraints large enough to encompass the to represent efforts to measure the Hubble constant, constraints from the two SNe groups. Perlmutter et pretation fits the data as well as or better ⍀ Ϫ ⍀ ϭϪ Ϯ but I also explore the h dependence of the age results al.(4) report 0.8( m) 0.6( ⌳) 0.2 0.1. ⌳ϭ0 than an explanation based on more myste- ⍀⌳ ϭ ⍀ ϭϪ Ϯ in Table 2. In the likelihood, LHubble, I distinguish the Evaluated at 0, this yields m 0.25 ៮ ⍀flat ϭ rious dark energy (4, 33, 34). observationally determined estimate h from the free 0.13. Assuming spatial flatness, they report m ϩ0.09(ϩ0.05) parameter h. 0.28Ϫ0.08(Ϫ0.04) statistical (systematic) errors. Add- 17. Y. Yoshii, T. Tsujimoto, K. Kawara, Astrophys. J. 507, ing the statistical and systematic errors in quadrature ⍀flat ϭ ϩ0.10 L113 (1998). yields m 0.28Ϫ0.09. The Riess et al.(5) con- References and Notes 18. M. Fukugita, C. J. Hogan, P. J. E. Peebles, ibid. 503, 518 straints are from their figures 6 and 7 (“MLCS meth- 1. I follow a common convention and work with dimen- (1998). od” and “snapshot method”), using either the solid or sionless quantities. The dimensionless Hubble constant is dotted contours, whichever is larger (corresponding ϭ Ϫ1 Ϫ1 19. Baryonic fraction of the mass of clusters of galaxies. h Ho/100 km s Mpc , where Ho is the usual Hubble ⍀ ⍀ ϭ Ϯ 4/3 Evrard (14) reports m/ b (12.5 0.7)h (in- to the analysis with and without SN1997ck, respec- constant in units of kilometers per second per megapar- ⍀ ⍀ ⍀ ϭ Ϯ ⍀⌳ϭ0 ϭϪ Ϯ ⍀flat ϭ dependent of ⌳), which becomes m/ b (12.5 tively). This yields m 0.35 0.18 and m ⍀ ⍀ ϩ0.17 sec. Both m and ⌳ are densities expressed in units of 4/3 0.24Ϫ . I use the weighted average of these Perl- ␳ ϭ 2 ␲ 3.8)h when the suggested 30% systematic error is 0.10 the critical density, which is defined as crit 3Ho/8 G, mutter et al.(4) and Riess et al.(5) constraints to ⍀ ϭ␳ ␳ added in quadrature. Burles and Tytler (15) report where G is Newton’s constant. Thus, m m/ crit, ⍀ 2 ϭ Ϯ obtain the SNe constraints listed above. The crucial ␳ bh 0.019 0.001, but Fukugita et al.(18) where m is the mass density of the universe in grams ⍀ 2 ϭ ⍀ favor a lower but larger range that includes bh upper error bars on m are large enough to include per cubic centimeter. The dimensionless cosmological ⍀ 2 ϭ Ϯ the constraints from either reference. ⍀ ϭ␳ ␳ ϭ⌳ 2 ⌳ 0.020. I adopt bh 0.015 0.005. When this constant ⌳ ⌳/ crit /3Ho, where is the cos- is combined with the Evrard (14) result, one gets the 36. One would like to be less Galactocentric, but mea- mological constant introduced in 1917 [A. Einstein, Sit- constraint ⍀ h2/3 ϭ 0.19 Ϯ 0.12. suring the ages of even nearby extragalactic objects zungsber. Preuss. Akad. Wiss. Phys. Math. KI. 1917, 142 m 20. The remaining five Li have no simple analytic form. is difficult. See K. A. Olsen et al.[Mon. Not. R. Astron. (1917); English translation in Principle of Relativity,H.A. 21. These combinations prefer the upper left region of Soc. 300, 665 (1998)], who find that the ages of old Lorentz et al., Eds. (Dover, New York, 1952), pp. 175]. the (⍀ , ⍀⌳) plane. I (9) have analyzed this comple- globular clusters in the Large Magellanic Cloud are 2. S. D. M. White, J. F. Navarro, A. E. Evrard, C. S. Frenk, m ⍀ ϭ Ϯ ⍀ ϭ mentarity and obtained ⌳ 0.62 0.16 and m the same age as the oldest Galactic globular clusters. Nature 366, 429 (1993). 0.24 Ϯ 0.10. The addition of the cluster abundance 37. D. B. Guenther and P. Demarque, Astrophys. J. 484, 3. C. H. Lineweaver and D. Barbosa, Astrophys. J. 496, evolution measurements and improvements in the 937 (1997). 624 (1998). SNe measurements tightens these limits to the val- 38. B. Chaboyer, E. M. Green, J. Liebert, available at 4. S. Perlmutter et al., ibid., in press (available at http:// ues given. http://xxx.lanl.gov/abs/astro-ph/9812097. xxx.lanl.gov/abs/astro-ph/9812133). 22. L(h, ⍀ , ⍀⌳) was derived as in (9). For a discussion 39. S. K. Leggett, M. T. Ruiz, P. Bergeron, Astrophys. J. flat ϭ m 5. A. G. Riess et al., Astron. J. 116, 1009 (1998). to of taking the maximum of the likelihood, rather than 497, 294 (1998). Ϯ 15.2 1.7 results from including a flat prior in the performing an to marginalize over the nui- 40. G. Carraro, A. Vallenari, L. Girardi, A. Richichi, Astron. analysis. sance variable, see (3, p. 626) and (11, Section 2.4) Astrophys. 343, 825 (1999) (available at http://xxx. 6. Galaxy cluster mass-to-light ratio limits are from R. and references therein. The lower right of Fig. 3A lanl.gov/abs/astro-ph/9812278). Carlberg et al.[Astrophys. J. 478, 462 (1997)] and R. shows that low h values are preferred (which corre- 41. R. L. Phelps, Astrophys. J. 483, 826 (1997). Carlberg et al. [paper presented at the 33rd Recontres ϰ spond to older ages because to 1/h). This prefer- 42. B. Chaboyer, P. Demarque, P. J. Kernan, L. M. Krauss, de Moriond, Fundamental Parameters in Cosmology, ence contrasts with the iso-to lines (Fig. 3A), which ibid. 494, 96 (1998). Les Arcs, France, 17 to 24 January, 1998 (available at have the younger ages in the lower right. Therefore, 43. R. G. Gratton et al., ibid. 491, 749 (1997). http://xxx.lanl.gov/abs/astro-ph/9804312)]. The error the three-dimensional CMB preferences will yield 44. I. N. Reid, Astron. J. 114, 161 (1997). bars in Table 1 are conservative in the sense that they tighter limits on to than is apparent from the figure. 45. M. Salaris and A. Weiss, Astron. Astrophys. 327, 107 include their “worst case,” 73% errors. 23. In the absence of any direct constraint on h, the (1997). 7. N. A. Bahcall and X. Fan, Astrophys. J. 504, 1 (1998); data are more consistent with large values of h. For 46. F. Grundahl, D. A. VandenBerg, M. T. Andersen, As- N. A. Bahcall, X. Fan, R. Cen, ibid. 485, L53 (1997). example, if no LHubble(h) is used in Eq. 1, the trophys. J. 500, L179 (1998). Table 1 lists their reported 95% errors, which I use as combined data yield a lower limit h Ͼ 0.67. The 47. J. J. Cowan et al., ibid. 480, 246 (1998). 1␴ error bars in this analysis. Ͻ corresponding age limit is to 14.2 Ga. No upper 48. R. Jimenez, available at http://xxx.lanl.gov/abs/astro- 8. E. J. Guerra, R. A. Daly, L. Wan, in preparation (avail- limit could be determined because the best fit for ph/9810311. able at http://xxx.lanl.gov/abs/astro-ph/9807249). I this data is h ϭ 1.00, at the edge of the range 49. F. Pont, M. Mayor, C. Turon, D. A. VandenBerg, Astron. have doubled the 1␴ errors quoted in the abstract. explored. This preference for higher h values (lower Astrophys. 329, 87 (1998). 9. C. H. Lineweaver, Astrophys. J. 505, L69 (1998). ages) helps explain the h dependence of the results 50. L. M. Krauss and G. D. Starkman, in preparation (avail- 10. M. White, ibid. 506, 495 (1998); A. M. Webster et al., in Table 2. Results for 0.60 Յ h Յ 0.80 are reported able at http://xxx.lanl.gov/abs/astro-ph/9902189). ibid. 509, 65 (1999); E. Gawiser and J. Silk, Science in Table 2. When a small uncertainty in h was 51. I thank R. De Propris for helpful discussions on the 280, 1405 (1998). assumed (effectively conditioning on an h value), it Galactic age estimates of Table 2. I acknowledge a 11. M. Tegmark, Astrophys. J. 514, L69 (1999). Gravita- dominated the h information from the other con- Vice-Chancellor’s Fellowship at the University of tional and effects were included in this straints. For example, although the central h values New South Wales. analysis weakening the CMB constraints on closed are the same, the results from using h ϭ 0.64 Ϯ ⍀ Ϫ⍀ ϭ Ϯ models (upper right of the m ⌳ plane). 0.02 and h 0.64 0.10 are quite different: 14.6 11 February 1999; accepted 4 May 1999

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