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References and Notes

  1. I follow a common convention and work with dimensionless quantities. The dimensionless Hubble constant is h = H0 / 100 km s-1 Mpc-1, where H0 is the usual Hubble constant in units of kilometers per second per megaparsec. Both Omegam and OmegaLambda are densities expressed in units of the critical density which is defined as rhocrit = 3H02 / 8pi G, where G is Newton's constant. Thus, Omegam = rhom / rhocrit where rhom is the mass density of the universe in grams per cubic centimeter. The dimensionless cosmological constant OmegaLambda = rhoLambda / rhocrit = Lambda / 3H02 where Lambda is the cosmological constant introduced in 1917 [A. Einstein, Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Kl., 142 (1917), English translation in Principle of Relativity, H.A. Lorentz et al. (Dover, New York, 1952) p. 175].

  2. S.D.M. White, J.F. Navarro, A.E. Evrard, C.S. Frenk, Nature 366, 429 (1993).

  3. C.H. Lineweaver and D. Barbosa, Astrophys. J. 496, 624 (1998).

  4. S. Perlmutter et al., ibid., in press, (available at

  5. A. G. Riess et al., Astron. J. 116, 1009 (1998). t0flat = 15.2 ± 1.7 results from including a flat prior in the analysis.

  6. Galaxy cluster mass-to-light ratio limits are from R. Carlberg et al., [Astrophys. J. 478, 462 (1997)] and R. Carlberg et al., [paper presented at the 33rd Rencontres de Moriond, Fundmental Parameters in Cosmology, Les Arcs, France, 17 to 24 January, 1998 (available at]. The error bars in Table 1 are conservative in the sense that they include their "worst case", 73% errors.

  7. N.A. Bahcall and X. Fan, Astrophys. J. 504, 1 (1998); N.A. Bahcall, X. Fan, R. Cen, ibid. 485, L53 (1997). Table 1 lists their reported 95% errors, which I use as 1sigma error bars in this analysis.

  8. E.J. Guerra, R.A. Daly, L. Wan, in preparation, (available at I have doubled the 1sigma errors quoted in the abstract.

  9. C.H. Lineweaver, Astrophys. J. 505, L69 (1998).

  10. M. White, ibid. 506, 495 (1998), A.M. Webster et al. ibid. 509, 65 (1998), E. Gawiser and J. Silk, Science 280, 1405 (1998).

  11. M. Tegmark, Astrophys. J. 514, L69 (1999). Gravitational and reionization effects were included in this analysis weakening the CMB constraints on closed models (upper right of the Omegam - OmegaLambda plane).

  12. G. Efstathiou, S.L. Bridle, A.N. Lasenby, M.P. Hobson, R.S. Ellis, Mon. Not. R. Astron. Soc. 303, 47 (1999).

  13. J.E. Felten and R. Isaacman, Rev. Mod. Phys. 58, 689 (1986). S.M. Carroll, W.H. Press, E.L. Turner, Annu. Rev. Astron. Astrophy. 30, 499 (1992) Eq. 17. There may be some ambiguity about what the "age of the universe" really means. I have used the Friedman equation derived from general relativity to extract the age of the universe. This age is the time that has elapsed since the early moments of the universe when the classical equations of general relativity became valid. In eternal inflation models [see, for example, A. Linde, Particle Physics and Inflationary Cosmology (Harwood Academic, Chur, Switzerland) p. 292ff (1990)] or other multiple universe scenarios ("multiverses"), the age of the universe is more complicated. If we live in a multiverse, then the age computed here refers only to the age of our bubble-like part of it.

  14. A.E. Evrard, Mon. Not. R. Astron. Soc. 292, 289 (1997).

  15. S. Burles and D. Tytler, Astrophys. J. 499, 699 (1998).

  16. B.F. Madore et al. (available at report h = 0.72 (± 0.05)r[± 0.12]s from an analysis which combines recent H0 measurements. The errors are random and systematic, respectively. F. Hoyle, G. Burbidge and J.V. Narlikar [Mon. Not. R. Astron. Soc., 286, 173, 1997] reviewed the observational literature and concluded h = 0.58+0.10-0.05. Both h = 0.65 ± 0.10 and h = 0.70 ± 0.10 are often cited as standard estimates [see for example (12, 18)]. I adopt h = 0.68 ± 0.10 to represent the world's efforts to measure the Hubble constant, but I also explore the h dependence of the age results in Table 2. In the likelihood, curlyLHubble, I distinguish the observationally determined estimate barh from the free parameter h.

  17. Y. Yoshii, T. Tsujimoto, K. Kawara, Astrophys. J. 507, L113 (1998).

  18. M. Fukugita, C.J. Hogan, P.J.E. Peebles, ibid. 503, 518 (1998).

  19. Baryonic fraction of the mass of clusters of galaxies. Evrard (14) reports Omegam / Omegab = (12.5 ± 0.7) h4/3 (independent of OmegaLambda) which becomes Omegam / Omegab= (12.5 ± 3.8) h4/3 when the suggested 30% systematic error is added in quadrature. Burles & Tytler (15) report Omegab h2 = 0.019 ± 0.001 but Fukugita et al. (18) favor a lower but larger range which includes Omegab h2 = 0.020. I adopt Omegab h2 = 0.015 ± 0.005. When this is combined with the Evrard (14) result, one gets the constraint: Omegam h2/3 = 0.19 ± 0.12.

  20. These five curlyLi have no simple analytic form.

  21. These combinations prefer the upper left region of the (Omegam, OmegaLambda) plane. I (9) have analyzed this complementarity and obtained OmegaLambda = 0.62 ± 0.16 and Omegam = 0.24 ± 0.10. The addition of the cluster abundance evolution measurements and improvements in the SNe measurements tightens these limits to the values given.

  22. curlyL(h, Omegam, OmegaLambda) was derived as in (9). For a discussion of taking the maximum of the likelihood, rather than doing an integral to marginalize over the nuisance variable, see (3, p. 626) and (11, Section 2.4) and references therein. The lower right of Fig. 3A shows that low h values are preferred (which correspond to older ages because t0 propto 1 / h). This preference contrasts with the iso-t0 lines (Fig. 3A) which have the younger ages in the lower right. Therefore, the three-dimensional CMB preferences will yield tighter limits on t0 than is apparent from the figure.

  23. In the absence of any direct constraint on h, the data are more consistent with large values of h. For example, if no curlyLHubble(h) is used in Eq. 1, the combined data yield a lower limit h > 0.67. The corresponding age limit is t0 < 14.2 Ga . No upper limit could be determined since the best fit for this data is h = 1.00, at the edge of the range explored. This preference for higher h values (lower ages) helps explain the h dependence of the results in Table 2. Results for 0.60 leq h leq 0.80 are reported in Table 2. When a small uncertainty in h was assumed (effectively conditioning on an h value), it dominated the h information from the other constraints. For example, although the central h values are the same, the results from using h = 0.64 ± 0.02 and h = 0.64 ± 0.10 are quite different: 14.6 and 13.5 Ga, respectively (Fig. 2). If t0 propto 1 / h were adhered to, there would be no difference.

  24. P. Stetson, D.A. VandenBerg, M. Bolte, Publ. Astron. Soc. Pac. 108, 560 (1996).

  25. T.D. Oswalt, J.A. Smith, M.A. Wood, P. Hintzen, Nature 382, 692 (1996); J.D. Oswalt, personal communication. Together, these references provided the age that is referred to as ``Oswalt98'' in Fig. 2.

  26. S.P. Driver, et al., Astrophys. J. 496, L93 (1998). But there is no consensus on this issue. See F.R. Marleau and L. Simard, ibid. 507, 585 (1998).

  27. E.E. Falco, C.S. Kochanek, J.A. Munoz, ibid. 494, 47 (1998).

  28. Y.N. Cheng and L.M. Krauss, in preparation (available at

  29. A.R. Cooray, in preparation, (available at

  30. R. Quast and P. Helbig, in preparation, (available at

  31. I. Zehavi, paper presented at the MPA/ESO Cosmology Conference, Evolution of Large-Scale Structure: From Recombination to Garching, 2 to 7 August 1998 (available at

  32. A. Dekel, D. Burstein, S.D.M. White, in Critical Dialogues in Cosmology N. Turok, Ed. (World Scientific, River Edge, NJ 1997), pp. 175-191.

  33. P.M. Garnavich et al., Astrophys. J. 509, 74 (1998).

  34. S. Perlmutter, M.S. Turner, M. White, available at

  35. I use SNe constraints large enough to encompass the constraints from the two SNe groups. Perlmutter etal. (4) report 0.8*Omegam - 0.6*OmegaLambda = -0.2 ± 0.1. Evaluated at OmegaLambda = 0 this yields OmegamLambda=0 = -0.25 ± 0.13. Assuming spatial flatness, they report Omegamflat = 0.28+0.09-0.08 (+0.05-0.04) statistical (systematic) errors respectively. Adding the statistical and systematic errors in quadrature yields Omegamflat = 0.28+0.10-0.09. The Riess etal. (5) constraints are from their figures 6 and 7 ("MLCS method" and "snapshot method") using either the solid or dotted contours whichever is larger (corresponding to the analysis with and without SN1997ck, respectively). This yields OmegamLambda=0 = -0.35 ± 0.18 and Omegamflat = 0.24+0.17-0.10. I use the weighted average of these Perlmutter etal. (4) and Riess etal. (5) constraints to obtain the SNe constraints listed above. The crucial upper error bars on Omegam are large enough to include the constraints from either reference.

  36. One would like to be less Galactocentric, but measuring the ages of even nearby extragalactic objects is difficult. See K.A. Olsen etal. [Mon. Not. R. Astron. Soc., 300, 665 (1998)] who find that the ages of old globular clusters in the Large Magellanic Cloud are the same age as the oldest Galactic globular clusters.

  37. D.B. Guenther and P. Demarque, Astrophys. J. 484, 937 (1997).

  38. B. Chaboyer, E.M. Green, J. Liebert, available at

  39. S.K. Leggett, M.T. Ruiz, P. Bergeron, Astrophys. J. 497, 294 (1998).

  40. G. Carraro, A. Vallenari, L. Girardi, A. Richichi, Astron. Astrophys. 343, 825, (1999). (available at

  41. 41. R.L. Phelps, Astrophys. J. 483, 826 (1997).

  42. 42. B. Chaboyer, P. Demarque, P.J. Kernan, L.M. Krauss, ibid. 494, 96 (1998).

  43. R.G. Gratton et al., ibid. 491, 749 (1997).

  44. I.N. Reid, Astron. J. 114, 161 (1997).

  45. M. Salaris and A. Weiss, Astron. Astrophys. 327, 107 (1997).

  46. F. Grundahl, D.A. VandenBerg and M.T. Andersen, Astrophys. J. 500, L179 (1998).

  47. J.J. Cowan et al., ibid. 480, 246 (1997).

  48. R. Jimenez, available at

  49. F. Pont, M. Mayor, C. Turon, D.A. VandenBerg, Astron. Astrophys. 329, 87 (1998).

  50. L.M. Krauss and G.D. Starkman, in preparation, (available at

  51. I thank R. De Propris for helpful discussions on the Galactic age estimates of Table 2. I acknowledge a Vice-Chancellor's Fellowship at the University of New South Wales.

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