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2. METHOD

Any measurement of a function of h, Omegam, and OmegaLambda can be included in a joint likelihood

Equation 1 (1)

which I take as the product of seven of the most recent independent cosmological constraints (Table 1 and Fig. 3). For example, one of the curlyLi in Eq. 1 represents the constraints on h. Recent measurements can be summarized as barh = 0.68 ± 0.10 (16). I represent these measurements in Eq. 1 by the likelihood,

Equation 2 (2)

Figure 3

Figure 3. The regions of the (Omegam, OmegaLambda) plane preferred by various constraints. (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 triangles (from upper left to lower right) in (A) are the approximate 1sigma, 2sigma and 3sigma confidence levels of the likelihood from CMB data, curlyLCMB (9). (A) also shows the important h dependence of curlyLCMB. The contours within the dark shaded region are of h values that maximize curlyLCMB for a given (Omegam, OmegaLambda) pair (h = 0.70, 0.90 contours are labeled). This correlation between preferred h and preferred (Omegam, OmegaLambda) helps curlyLCMB(h, Omegam, OmegaLambda) constrain t0. In (B) through (E), thin contours enclose the 1sigma (shaded) and 2sigma confidence regions from separate constraints, and thick contours indicate the 1sigma, 2sigma and 3sigma regions of the combination of curlyLCMB 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) (thick contours). The best fit values are OmegaLambda = 0.65 ± 0.13 and Omegam = 0.23 ± 0.08. In (A), the thin iso-t0 contours (labeled ``10'' through ``14'') indicate the age in Ga when h = 0.68 is assumed. For reference, the 13- and 14-Ga contours are in all the 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 = 0.58 and h = 0.78 (the 1sigma limits of the principle h estimate used in this paper). In (F), it appears that the best fit has t0 approx 14.5 Ga, but all constraints shown here are independent of information about h; they do not include the h dependence of curlyLCMB, curlyLbaryons or curlyLHubble (Table 1).

Table 1: Parameter estimates from non-CMB measurements. I use the error bars cited here as 1sigma errors in the likelihood analysis. The first four constraints are plotted in Fig. 3 B through E.
Method Reference Estimate
SNe (35) OmegamLambda=0 = -0.28 ± 0.16 Omegamflat = 0.27 ± 0.14
Cluster mass-to-light (6) OmegamLambda=0 = 0.19 ± 0.14
Cluster abundance evolution (7) OmegamLambda=0 = 0.170.28-0.10 Omegamflat = 0.22+0.25-0.10
Double radio sources (8) OmegamLambda=0 = -0.25+0.70-0.50 Omegamflat = 0.1+0.50-0.20
Baryons (19) Omegam h2/3 = 0.19 ± 0.12
Hubble (16) h = 0.68 ± 0.10

Another curlyLi in Eq. 1 comes from measurements of the fraction of normal baryonic matter in clusters of galaxies (14) and estimates of the density of normal baryonic matter in the universe [Omegabh2 = 0.015 ± 0.005 (15, 18)]. When combined, these measurements yield omhbar = 0.19 ± 0.12 (19), which contributes to the likelihood through

Equation 3 (3)

The (Omegam, OmegaLambda)-dependencies of the remaining five constraints are plotted in Fig. 3 (20). The 68% confidence level regions derived from CMB and SNe (Fig. 3, A and B) are nearly orthogonal, and the region of overlap is relatively small. Similar complementarity exists between the CMB and the other data sets (Figs. 3, C through E). The combination of them all (Fig. 3F) yields OmegaLambda = 0.65 ± 0.13 and Omegam = 0.23 ± 0.08 (21).

This complementarity is even more important (but more difficult to visualize) in three-dimensional parameter space: (h, Omegam, OmegaLambda). Although the CMB alone cannot tightly constrain any of these parameters, it does have a strong preference in the three-dimensional space (h, Omegam, OmegaLambda). In Eq. 1, I used curlyLCMB(h, Omegam, OmegaLambda), which is a generalization of curlyLCMB(Omegam, OmegaLambda) (Fig. 3A) (22). To convert the three-dimensional likelihood curlyL(h, Omegam, OmegaLambda) of Eq. 1 into an estimate of the age of the universe and into a more easily visualized two-dimensional likelihood, curlyL(h, t0), I computed the dynamic age corresponding to each point in the three-dimensional space (h, Omegam, OmegaLambda). For a given h and t0, I then set curlyL(h, t0) equal to the maximum value of curlyL(h, Omegam, OmegaLambda)

Equation 4 (4)

This has the advantage of explicitly displaying the h dependence of the t result. The joint likelihood curlyL(h, t0) of Eq. 4 yields an age for the universe: t0 = 13.4 ± 1.6 Ga (Fig. 4). This result is a billion years younger than other recent age estimates.

Figure 4

Figure 4. This plot shows the region of the h - t0 plane preferred by the combination of all seven constraints. The result, t0 = 13.4 ± 1.6 Ga, is the main result of this paper. The thick contours around the best fit (indicated by a star) are at likelihood levels defined by curlyL / curlyLmax = 0.607, and 0.135, which approximate 68% and 95% confidence levels, respectively. These contours can be projected onto the t0 axis to yield the age result. This age result is robust to variations in the Hubble constraint as indicated in Table 2. The areas marked ``Excluded'' (here and in Fig. 5) result from the range of parameters considered: 0.1 leq Omegam leq 1.0, 0 leq OmegaLambda leq 0.9 with Omegam + OmegaLambda leq 1. Thus, the upper (high t0) boundary is defined by (Omegam, OmegaLambda) = (0.1, 0.9), and the lower boundary is the standard Einstein-deSitter model defined by (Omegam, OmegaLambda) = (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 Milky Way (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 = 0.73 ± 0.09, a slightly higher and tighter estimate than the input h = 0.68 ± 0.10).

What one uses for curlyLHubble(h) in Eq. 1 is particularly important because, in general, we expect the higher h values to yield younger ages. Table 2 contains results from a variety of h estimates, assuming various central values and various uncertainties around these values. The main result t = 13.4 ± 1.6 Ga has used h = 0.68 ± 0.10 but does not depend strongly on the central value assumed for Hubble's constant (as long as this central value is in the most accepted range, 0.64 leq h leq 0.72) or on the uncertainty of h (unless this uncertainty is taken to be very small). Assuming an uncertainty of 0.10, age estimates from using h = 0.64, 0.68 and 0.72 are 13.5, 13.4 and 13.3 Ga, respectively (Fig. 2). Using a larger uncertainty of 0.15 with the same h values does not substantially change the results, which are 13.4, 13.3, 13.2 Ga, respectively. For both groups, the age difference is only 0.2 Gy. If t0 propto 1 / h were adhered to, this age difference would be 1.6 Gy. Outside the most accepted range the h dependence becomes stronger and approaches t0 propto 1 / h (23).

Figure 5

Figure 5. 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 = 0.68 ± 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.

To show how each constraint contributes to the result, I convolved each constraint separately with Eq. 2 (Fig. 5). The result does not depend strongly on any one of the constraints (see ``all - x'' results in Table 2). For example, the age, independent of the SNe data, is t0(all - SNe) = 13.3+1.7-1.8 Ga, which differs negligibly from the main result. The age, independent of the SNe and CMB data, is 0(all - CMB - SNe) = 12.6+3.4-2.0 Ga, which is somewhat lower than the main result but within the error bars.

Table 2: Age estimates of our Galaxy and universe (36). ``Technique'' refers to the method used to make the age estimate. OC, open clusters; WD, white dwarfs; LF, luminosity function; GC, globular clusters; M/L, mass-to-light ratio; and cl evol, cluster abundance evolution. The averages are inverse variance-weighted averages of the individual measurements. The sun is not included in the disk average. ``Isotopes'' refers to the use of relative isotopic abundances of long-lived species as indicated by absorption lines in spectra of old disk stars. The ``stellar ages'' technique uses main sequence fitting and the new Hipparcos subdwarf calibration. ``All'' means that all six constraints in Table 1 and the CMB constraints were used in Eq. 1. ``All-x'' means that all seven constraints except constraint x were used in Eq. 1. Figures 3 and 5 and the all - x results indicate a high level of agreement between constraints and the lack of dependence on any single constraint. Thus, there is a broad consistency between the ages preferred by the CMB and the six other independent constraints. Figure 2 presents all of the disk and halo age estimates.
Technique Reference h Assumptions Age (Ga) Object
Isotopes (37) None 4.53 ± 0.04 Sun
Stellar ages (38) None 8.0 ± 0.5 Disk OC
WD LF (39) None 8.0 ± 1.5 Disk WD
Stellar ages (40) None 9.0 ± 1 Disk OC
WD LF (25) None 9.7+0.9-0.8 Disk DW
Stellar ages (41) None 12.0+1.0-2.0 Disk OC
None 8.7 ± 0.4 tdisk(avg)
Stellar ages (42) None 11.5 ± 1.3 Halo GC
Stellar ages (43) None 11.8+1.1-1.3 Halo GC
Stellar ages (44) None 12 ± 1 Halo GC
Stellar ages (45) None 12 ± 1 Halo GC
Stellar ages (46) None 12.5 ± 1.5 Halo GC
Isotopes (47) None 13.0 ± 5 Halo stars
Stellar ages (48) None 13.5 ± 2 Halo GC
Stellar ages (49) None 14.0+2.3-1.6 Halo GC
None 12.2 ± 0.5 tGal (avg)
SNe (4) 0.63 ± 0.0 14.5 ± 1.0 Universe
SNe (flat) (4) 0.63 ± 0.0 14.9+1.4-1.1* Universe
SNe (5) 0.65 ± 0.02 14.2 ± 1.7 Universe
SNe (flat) (5) 0.65 ± 0.02 15.2 ± 1.7* Universe
All This work 0.60 ± 0.10 15.5+2.3-2.8 Universe
All This work 0.64 ± 0.10 13.5+3.5-2.2* Universe
All This work 0.68 ± 0.10 13.4+1.6-1.6* Universe
All This work 0.72 ± 0.10 13.3+1.2-1.9* Universe
All This work 0.76 ± 0.10 12.3+1.9-1.6 Universe
All This work 0.80 ± 0.10 11.9+1.9-1.6 Universe
All This work 0.64 ± 0.02 14.6+1.6-1.1* Universe
All - CMB This work 0.68 ± 0.10 14.0+3.0-2.2 Universe
All - SNe This work 0.68 ± 0.10 13.3+1.7-1.8 Universe
All - M/L This work 0.68 ± 0.10 13.3+1.9-1.7 Universe
All - cl evol This work 0.68 ± 0.10 13.3+1.7-1.4 Universe
All - radio This work 0.68 ± 0.10 13.3+1.7-1.5 Universe
All - baryons This work 0.68 ± 0.10 13.4+2.6-1.5 Universe
All - Hubble This work None < 14.2 Universe
All - CMB - SNe This work 0.68 ± 0.10 12.6+3.4-2.0 Universe
* Also plotted in Fig. 2.

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