The concept of time has always held a special fascination for humans. Even before Einstein provided us with Special Relativity and its notions of space-time, people mused on the possibility of knowing the future. Those who purported to be able to foretell events were often confidants of the power brokers of the day. Even more intriguing was the notion that one could somehow move through time in the way we move through space, both forward and backward. Science fiction contains many tales anchored solidly in this notion.
The passing of time is of particular significance to people. We annually mark the time since our birth with a special day, and when a sufficient number of these celebrations have occurred we are allowed to participate in certain rituals of our culture. For example, after 16 celebrations, we are permitted to terrorize fellow citizens and cause considerable expense to our parents by driving a motor vehicle. After 35 celebrations we are deemed capable of holding the highest political office in the land!
Viewed quantitatively, the rate at which we move with the arrow of time is well determined and specific (currently clocks are available that are accurate to one part in 1,000,000,000,000,000!). However, our qualitative sense of this rate of passage is another matter. Thomas Mann in his classic work Magic Mountain explores the fact that humans perceive the passage of time in ways that are extremely subjective. As we age, our sense of the passage of time leads to an often alarming compression of the interval between events. We all recall that when in elementary school the few months of summer seemed to last an eternity. Yet most of us now are astonished to find that it is time to once again write our annual progress report. Didn't we just submit the damn thing?!
It is the issue of age, and how we determine the passage of time, that is the subject of this article. I am here concerned not with the age of any old system or individual, but rather the age of the universe. The motivation for this brief contribution is the interest and confusion, both in the public and among scientists, generated by the announcement in the fall of 1994 that a team of astronomers had observed Cepheid variable stars in a distant galaxy using the refurbished Hubble Space Telescope. Based on those observations one could conclude that the universe was only about 8 billion years old. While that was not cause for concern for many people, others immediately recognized that this age was in conflict with the generally quoted age of the galaxy based on the oldest stars it contains: If the interpretation by the Hubble team was correct, the galaxy was older than the entire universe! Something was amiss.
A member of the Institute staff, during a chance encounter in the hall, asked me to explain this conflict. I faked it at the time. That experience and discussions with colleagues during the most recent Lunar and Planetary Science Conference led me to examine this issue more carefully to better understand the significance of the Hubble observations. What follows are the results of an admittedly brief effort that served as the foundation for a seminar that I gave recently at the Institute. The editor of the Bulletin then bullied me into summarizing the seminar for publication here.
Any discussion requires a shared language or concept. It is in that spirit that I present a Readers Digest version of the history of the universe in order to define the context of this paper as well as some terminology.
The chronology of the universe is depicted schematically in Fig. 1. While some remarkable events occurred in the earliest phases of the evolution of the universe, the intervals of time that are relevant to our discussion are those indicated by the symbols and . The first of these is the interval of time since the Big Bang until the epic of galaxy formation (we are actually interested only in the time when our galaxy was formed, but we will use a number that is representative of many galaxies). The second interval marks the passage of time from the formation of the galaxy until the solar system was formed. The third interval corresponds to the age of the solar system. A value of 1.0�0.4 billion years will be taken here for the interval
These intervals are relevant to two approaches to establishing the age of the universe, both involving ways to estimate the age of the galaxy. One way is to estimate the age of the oldest star(s) in the galaxy and associate that with the age of the galaxy. The other way has to do with the age of the elements, the majority of which were created by stellar processes in the galaxy.
The intervals shown in Fig. 1 are not germane to the method for estimating the age of the universe using the recent Hubble observations. Ignoring for the moment subtle, but fundamental, nuances regarding the early history of the universe, its current expansion is well described by what is known as the Friedmann equation:
where H(t) is the Hubble constant at time t, T(t) is the Hubble time, a(t) is an arbitrary distance metric of the universe, k is Einstein's curvature parameter, and and are the mass-energy density of matter and radiation and the vacuum mass-energy density respectively. A universe that is flat and Euclidean in three dimensions is described by
This equation is often expressed in terms of the dimensionless quantities and where
The subscript denotes values at the current epoch. The age of the universe, , is some multiple of the Hubble time, , with the multiple depending on the exact values of , , and the so-called deceleration parameter, , where
A standard Einstein-de Sitter model of the universe with = 1 yields = (2/3). The variation of with and is shown in Fig. 2. We will return to Fig. 2 later in this article.
Fig. 2. Current limits on and the age and density of the universe. Expansion ages of 14�2 Gyr, corresponding to globular cluster ages, are illustrated as a function of and the density parameter for models with the cosmological constant . The , , and confidence intervals on the Hubble constant presented here are shown by the lightly hatched, densely hatched, and black horizontal areas respectively. Broad limits on are illustrated at 0.1 and 1.0.
Different workers approach the problem in different ways. Many researchers consider only the systematics of element creation in stars (nucleosynthesis) in their approach to estimating the age of the universe (e.g., papers by Fowler and coworkers).
Others argue that neucleosynthesis alone cannot give us a firm value for the age of the galaxy. These workers (e.g., papers by Clayton and coworkers) argue that one needs to consider the dynamical history of the galaxy, with emphasis on extragalactic material that falls into the galaxy over time ("infall"), as well as the record revealed by radioactive decay over the age of the galaxy.
Space does not permit a full review of this important area of research, so I will only give the basic results from the published literature. Interested readers can refer to the review by Cowan et al. (1991) and references therein for more details.
One thing that is agreed upon by all workers in the field is that the age of the solar system,, is 4.6�0.1 billion years. The trick is to obtain a value for the parameter , the time between the formation of the galaxy and the formation of the solar system.
One can obtain a simple lower limit for from the following argument. If we assume that all of the elements, but notably uranium, were created in a single event prior to formation of the solar system, then
where P and A are respectively the production and abundance ratios of a radioactive, R, and a "stable," S, isotope. Using and as the radioactive and "stable" isotopes respectively, and values of P = 1.34 and A = 0.283 from Fowler, we find that
1.87 billion years
Taken in conjunction with the age of the solar system, we find a lower limit to the age of the galaxy of 6.5 billion years and the age of the universe of 7.5 billion years.
Fowler, in a series of papers at the end of the last decade, argues that the available data suggest that the best value for the parameter is 5.4�1.5 billion years. This value, taken with the other time interval values, yields an age of the universe = (1.0 + 5.4 + 4.6) = 11.0�1.6 billion years.
Clayton and coworkers employ a different approach to estimating the age of the elements through the development of a family of analytic models of the chemical evolution of the galaxy (Clayton, 1989). He finds that in the case of a closed galaxy, i.e., one that is not affected by addition of material over its history, the chronometer based on the isotopes and gives
5.8 billion years
A similar estimate, based on the isotope pair and , yields
9.4 billion years
The former estimate by Clayton gives 11.4 billion years, while the latter estimate gives < 20 billion years, or 13 < < 21 billion years. A nearly identical range in ages is argued by Cowan et al. (1991).
The fact that stars tend to be formed in large family units, as opposed to birth in obvious isolation from other stars, has been a boon to our understanding of many aspects of star formation and evolution. Notable among these family units are galactic clusters, and in particular globular clusters. What makes globular clusters, especially the metal-poor ones, so important in this discussion is that the place on the HR diagram where the cluster "turns off" from the main sequence curve provides a measure of the age of that cluster (Fig. 3).
A recent study of globular clusters by Chaboyer (1995) indicates that the apparent age spread of clusters is more likely due to our general lack of observational knowledge about certain regions of the HR diagram, rather than flaws in our detailed models of stars. Chaboyer provides an informative and extensive assessment of this issue and concludes that the ages of the oldest galactic globular clusters is in the range 13-17 billion years, giving 14 < < 18 billion years. A similar conclusion is reached by van den Bergh (preprint to a paper presented at the Dahlem Workshop in 1995) who concluded that globular clusters indicate an age for the galaxy of 18 billion years, or = 19 billion years.
Fig. 3. A composite color versus magnitude diagram of 10 galactic clusters and 1 globular cluster. Ages corresponding to the various main-sequence termination points are given along the righthand ordinate. The zero-age main sequence is taken to be the blue envelope of the observed sets of main-sequence stars. Notice the rapidly evolved red giants in h +x Persei, which are apparently no more than 2 million years old. Some white dwarfs are known in the Hyades, indicating that it is possible to form them in a few million years, either directly or as the end product of the evolution of upper-main-sequence stars. Curiously enough, the Hyades has no red giants. The oldest galactic cluster, M 67, is older than the Sun and has scores of white dwarfs. Many fascinationg problems are uncovered in the attempts to interpret the star densities in these diagrams quantitatively. [After A. Sandage, Astrophys. J., 125:435 (1957). By permission of The University of Chicago Press. Copyright 1957 by The University of Chicago.]
We return now to the observations that caused the furor. Shown in Fig. 4 is a Hubble Telescope image of the galaxy M100, located in the Virgo cluster of galaxies. It is in this galaxy that a team of astronomers detected Cepheid variable stars using the Hubble Telescope. A no-less-spectacular, but less-well- publicized, set of observations by a team using the groundbased Canada-France-Hawaii Telescope (CFH) also detected Cepheid variables, but in the galaxy NGC 4571. This galaxy, like M100, is located in the Virgo cluster. The characteristic luminosity changes of a Cepheid variable in M100 are shown in Figs. 5 and 6.
Fig. 4. Hubble image of the galaxy M100.
Figs. 5 and 6. Hubble Telescope images taken with the Wide Field Planetary Camera 2 of Cepheid variable stars in M100. The Cepheid (center of each image frame) exhibits the characteristic, periodic change in luminosity that allows astronomers to use them as "standard candles" in estimating distances in the universe.
Cepheid variables are giant stars that exhibit regular, periodic dimming and brightening. They are of great value in establishing distances because they provide a means of assessing their intrinsic brightness through the Cepheid period-luminosity relation. If one can determine the period of the variability, one can determine the intrinsic brightness of the star. Given the intrinsic brightness and the apparent brightness, the Cepheid's distance can then be determined. There are many subtle aspects of this logic chain, but suffice it to say that, when due care was taken, these two groups of observers obtained distances to the Virgo cluster that were in reasonable agreement with one another.
The Hubble team concluded that M100 lies some 17.1 megaparsecs (Mpc) away, while the CFH team concluded that NGC 4571 is at a distance of 14.9 megaparsecs. While a difference of just over 2 megaparsecs may seem like a lot to people who do precise isotopic analyses, it should be pointed out that the Virgo cluster contains some 2000 galaxies and a separation of 2 megaparsecs between members of the cluster would not be a surprise.
The Hubble constant given by the CFH observations is whereas the corresponding value from the Hubble observations is .
While these new results are significant because they derive from measurements involving more distant galaxies than ever before, the inferred value of is in keeping with values obtained previously. According to van den Bergh (1994) the preferred indicators from all data prior to 1994 give a value of . The key word here is "preferred," as there is a large body of evidence that is consistent with . Typically, these latter data involve distance measures using supernovae as standard candles. An interesting variant on using supernova light curves that has been advanced recently by Reiss et al. (1995) yields a value of .
The wide variation in recently reported values of the Hubble constant must be a concern for any serious student of this subject. Until there is an adequate explanation of why this discrepancy exists, it will be difficult to take too seriously any inferences regarding the age of the universe based on Hubble Constant values alone.
The situation is complicated further by the fact that even if an agreed-to value for the Hubble constant were available, there remains uncertainty as to how that relates to the true age of the universe. Recall that the inverse of the Hubble constant gives the Hubble time, it does not give the age of the universe. In order to obtain the age of the universe from the Hubble time there must be additional information regarding the parameters discussed in connection with equation (3).
Figure 2 is a plot depicting how given values of and combine to yield an age of the universe. The white area in the figure indicates the nominal value and uncertainty in the Hubble constant obtained by the Hubble team taken in conjunction with a possible range in the value of from 0.1 to 1.0. As can be seen, this box intersects the line labeled "12 Gyr" as the age of the universe for low values of . Perhaps more significant is that the limits on the Hubble measurement are easily consistent with ages of the universe that are near or slightly in excess of 15 billion years.
My personal view is that the current data are not totally inconsistent with one another when all the uncertainties inherent in each of the three methods are considered. If pushed, I would bet on the galactic measures, viz., cosmochronology and the ages of the oldest galactic clusters, as giving the more accurate measures of the age of the universe. They are two independent methods and they give results that are broadly in agreement with one another. That there remains a significant issue in the value of the Hubble constant, along with uncertainties in relating that constant to the age of the universe, indicates that one should not be overly enthusiastic about determining the age of the universe through measures of the Hubble constant alone.
The debate about the age of the universe will certainly continue, but let us hope that the quality of the debate is not characterized by the following quotation:
"BRICK: Well, they say nature hates a vacuum, Big Daddy. BIG DADDY: That's what they say, but sometimes I think that a vacuum is a hell of a lot better than some of the stuff that nature replaces it with.
(Tennessee Williams, Cat on a Hot Tin Roof, Act 2).
References: Chaboyer B. (1995) Astrophysical Journal, 444, L9-12. Clayton D. D. (1989) 14th Texas Symposium on Relativistic Astrophysics (E. J. Fenyores, ed.), pp. 79-89, N.Y. Acad. Sci., New York. Cowan J. J., Thielemann F. K., and Truman J. W. (1991) Annu. Rev. Astron. Astrophys, 29, 447-497. Fowler W. (1989) 14th Texas Symposium on Relativistic Astrophysics (E. J. Fenyores, ed.), pp. 68-78, N.Y. Acad. Sci., New York. Riess A. G., Press W. H., and Kirshner R. P. (1995) Astrophysical Journal, 438, L17-20. van den Bergh S. (preprint) The Age of the Universe.
(David C. Black is the director of LPI)