Determination of the Universe's Age, to

Daniel Perley

Table of Contents Introduction

One of the simplest, and most scientifically important, questions one can ask is the age of the universe we inhabit. Under Big Bang cosmology this value, "to", simply refers to the time elapsed since the Big Bang itself, though in fact it could just as well apply (and be measured) had Big Bang model proved incorrect.

The central importance of the determination of this value is equally clear: as although science is in part a practical endeavor, it is also uniquely situated to provide answers to grand questions about the history and mechanics of the universe itself. The question of the universe's antiquity is in particular one of the oldest questions humans have pondered, and its importance understood by nearly everyone.

Still, our objectives in determining the age of the universe need not be so abstract and philosophical. In addition to being an interesting scientific objective in its own right, measurements of and constraints on the age of the universe provide valuable insight into the accuracy of our cosmological models as a whole for example, direct dating of the oldest clusters and stars called into question the favored notion that the universe's expansion was really slowing down with time long before supernova redshifts demonstrated that the universe is in fact expanding. Its importance in this sense extends even beyond cosmology to questions about all aspects of astronomy and science the time available since the universe's formation constrains models of the formation of stars and galaxies, the enrichment of the universe with heavy elements, and perhaps the development of life.


Cosmological Methods

Early in the 20th century, Edwin Hubble made the momentous discovery that the universe was apparently expanding distant galaxies were receding from our own, and in fact the speed of a galaxy's recession was proportional to the distance. While this discovery was surprising enough on its own (the expectation before then had been of a universe that was largely static on very large scales), equally important were the implications about the universe. If galaxies are flying apart from each other now, then in the past the galaxies were much closer together and at some point far enough into the past were concentrated into a region of effectively zero volume, clearly suggesting a finite age of the universe, not to mention a fiery birth from a region of extreme density and temperature now known as the 'Big Bang'. Furthermore, the timescale for this process follows quite naturally from the linear relationship observed by Hubble simply divide the distance between any two galaxies by the recession speed to calculate the time elapsed since they were together in the Big Bang. The linear nature of the expansion ensures that this value will be the same for any pair of galaxies.

This simplistic calculation, of course, assumes that the Hubble constant has not changed in time; one cannot even completely rule out that at some point in the past the universe was not expanding at all and the Big Bang never happened. Some early 20th century scientists uneasy with the Big Bang proposed an alternative theory, also compatible with Hubble's observations, that galaxies are continually created as the universe expands, allowing the universe's age to be infinite, calling out the "Steady State Theory". However, the discovery (and later measurement in detail) of the cosmic microwave background radiation (CMB), the relic radiation from the hot, dense early universe following the Big Bang, has validated the Big Bang model almost beyond doubt, as have various other cosmological measurements.

Figure 1 - Evolution of the size of the universe (scale factor) with time under various matter-dominated models. The time before present at which the size goes to zero indicates the universe's age to.
Nevertheless, while the conclusion that the observable universe was once concentrated into an arbitrarily small volume appears to be correct, there is good reason to believe that the expansion is not constant with time. The universe contains matter, and with matter comes gravity, which exerts a force pulling the matter in the universe towards itself and slowing the expansion as time progresses. Looking backward into time, this means that since the expansion has been slowing down due to the influence of gravity, the expansion was actually faster in the past making the "real" age of the universe younger than the direct extrapolation method would suggest, by an amount determined by the density of matter in the universe, as shown in Figure 1.

Due to difficulties in measuring the distances to faraway objects, the exact age determined by the Hubble expansion has been quite uncertain until fairly recently. With the currently accepted value for the Hubble constant Ho of 71 km/s/Mpc, a universe with no gravity would have an age of 13.7 billion years. Taking into account the decelerating effect of gravity using current measurements of the universe's mass density (including both normal "baryonic" matter in stars and gas plus the mysterious dark matter that binds galaxies and clusters together), this shortens to about 11 billion years, and using the matter density predicted by a successful specific Big Bang theory known as inflation, this shortens to only about 9 billion years.

Figure 2 - According to recent measurements, the universe contains a mysterious 'dark energy' in addition to normal matter, which causes the expansion speed to accelerate. The age of the universe depends on the amount matter and the amount of dark energy. This contour plot shows the age derived from different possible values of the current matter density (x-axis) and dark energy density (y-axis); the actual values are thought to be 0.3 (matter) and 0.7 (dark energy), corresponding to the red dot.
None of these values are correct, however, and in fact the latter two are seriously in conflict with the measured ages of the oldest stars and clusters (to be discussed later in this report) of about 12-13 billion years. In 1998, astronomers studying distant supernovas (which, by virtue of their immense distances, effectively sample the recession velocity of the universe in the very distant pasts when their light was emitted) determined that the universe was not slowing down with time, but in fact speeding up at least in the recent past. While the reason remains somewhat of a mystery, the entity responsible for this acceleration has been called 'dark energy' and functions as an effective antigravity. More recently, more detailed measurements of supernovae as well as analysis of the cosmic microwave background have measured the effect of this expansion in detail, and factoring it into the equation for the age of the universe, our best cosmological model (corresponding to the red dot in the contour plot of Figure 2) indicates an age of 13.7 +/- 0.2 billion years since the Big Bang.


Despite our high degree of confidence in the Big Bang model, it is important to provide independent checks on its predictions in this case, the estimate of the universe's age t0. Fundamentally this is quite simple: determine the age of an object in the universe whose age we think we know to reasonable accuracy; that age then sets a very strict minimum for the age of the universe. (It is, after all, hard to see how an object within the universe could be older than the universe itself.) If we add in our knowledge of conditions shortly after the Big Bang, we can even do slightly better the first stars, clusters, and galaxies were all thought to have formed within a few hundred million years after the Big Bang, so a rough estimate of the universe's age can also be made directly.

These measured ages can then be used as a consistency check on the accuracy of our overall theory if the ages agree, our confidence in the model increases; if they do not (and especially if the ages of measured objects are too old), then our theory is in error and needs modification. While today the observed ages are in good agreement with theory, historically, before the cosmological parameters were as well-known as they are today, the ages provided very important constraints: ages of 14 billion years for the oldest known stars and star clusters appeared highly inconsistent with the cosmological estimates of 9-11 billion years, suggesting problems with the current cosmological paradigm that were only truly solved by the discovery of dark energy and the accelerating universe. (Realization of measurement errors that made the observed ages just slightly too old also contributed to alleviating the paradox.)

Of course, measuring the ages of objects themselves, especially very old ones, is decidedly non-trivial. The challenge is to find an object which behaves with extreme regularity with minimal dependence on its environment, and yet does change its properties with time in some measurable way a sort of "cosmic clock".

As it turns out, there are two such classes of clock that have been widely used for dating the age of the universe, which operate on about the most radically different scales imagineable: radioactive nuclei, and stars. The two methods (and, in fact, several sub-methods within each method) are quite independent, and provide mutually supportive estimates of the age of the universe.


Radiometric Methods

Figure 3 - The exponential decay curve. The concentration remaining decreases by a factor of two every half-life.
Radioactive elements are inherently unstable, and any radioactive atom will eventually (given enough time) eject a particle of some sort and decay into a different element or isotope. While for a single atom this decay is essentially random and unpredictable, for a a large enough ensemble of atoms of the same radioactive isotope, this random behavior will statistically average out and the ensemble as a whole will behave in a well-defined manner: the fraction of atoms that decays will be a constant with time (so if 10% of the atoms decay in the first hour, 10% of the remaining atoms will decay the second hour.) This leads to the familiar exponential decay curve, plotted at right (Figure 3) and given by the function

N(t) = Ni 2-t/t1/2

or

N(t) = Ni e-t/[t1/2 / ln2]

Where t indicates the time elapsed since the sample had an abundance for the isotope of Ni. The term t1/2 is the half-life of the isotope, which indicates the amount of time required for half of the sample to decay. Inverting this equation gives an obvious way to determine the time elapsed, t:

t = -(t1/2 / ln2) ln (N/Ni)

The half-life is a constant for each isotope and is easily measured in the laboratory. The current concentration can similiarly measured by laboratory methods, or by spectroscopy (as will be discussed later.) However, measuring the initial concentration can be tricky.


Earth and the Solar System
Isotopet1/2 (years)
Samarium-146103,000,000
Uranium-236 234,200,000
Uranium-235 703,800,000
Potassium-40 1,280,000,000
Uranium-238 4,468,000,000
Rubidium-87 4,750,000,000
Thorium-232 14,100,000,000
Lutetium-176 37,800,000,000
Rhenium-187 43,500,000,000
Lanthanium-138 105,000,000,000
Samarium-147 106,000,000,000
Table 1 - Half-lives of various long-lived radioactive isotopes.

Only a few isotopes are suitable for long-term radiometric dating. If the half-life for the isotope is much longer than the age of the object it contains, the amount of decay will be very small, and the error in the derived age will be comparable to the age itself, a fairly useless result. If the half-life is much shorter (by a factor of more than about 10), then due to the exponential nature of the decay the element will decay to undetectable levels. Ideally the half-life of the element studied should be comparable to its anticipated age (within a factor of 2 or 3).


A list of isotopes of appropriate half-lives for a universe of anticipated age is presented in Table 1. Many of these objects are in fact used in long-term dating; specifically, rhenium-187, thorium-232, and uranium-238 have all been used to date the universe.

By far the most accurate radiometric dates come from samples from Earth (and elsewhere in the Solar System) analyzed in labs. In these cases, scientists can not only measure the amount of radioactive element to extremely high precision; in addition, they can also measure quite accurately the amount of decay product that accumulates, since in general it remains trapped in the solid rock which can be used to determine the initial concentration to quite high accuracy. The Earth's age of 4.55 billion years is known with an error of less than one percent from the dating of meteorites using a variety of elements from the table above. (Meteorites are used because they all formed very early in the history of the Solar System, whereas plate tectonics constantly creates and destroys the surface rocks on Earth; Earth rocks can provide only a minimum age.)

Figure 4 - A typical stellar spectrum, with absorption lines of several ions prominent. The concentration of these ions can be measured by the area of their lines - one might estimate the carbon content of this star by the amount of area shaded in red.
Old stars
Of course, while Earth's age provides a very strict lower limit on the age of the universe due to the high accuracy of the observations, it is well below that predicted by any mainstream cosmological model and is not a significant constraint on cosmological theory. However, the same method can be applied to objects well outside the solar system using spectroscopy. High-resolution measurements of the spectra of stars can reveal the presence of even trace elements and isotopes including, in some cases, the rare radioactive isotopes mentioned above as absorption lines. The total width of these absorption lines indicates (if the temperature, surface gravity, and other parameters of the star are known) the concentration in the stellar atmosphere (Figure 4).

Unfortunately there is no direct way of inferring the initial concentration of these elements. However, the concentration can be estimated indirectly using our understanding of the initial generation of heavy elements (both radioactive and nonradioactive) in supernovae. Supernovae, uncontrolled, explosions of massive stars are though to produce all elements in the universe heavier than iron (and most elements lighter than iron except for hydrogen and helium as well); furthermore, nuclear physics predicts the relative concentration of various elements that will be produced during the supernova; predictions that have been largely vindicated by observations of nearby supernova.
Figure 5 - Theoretical (line) and observed (data points) abundances for various heavy elements in an old star. All the points match up with theory - except the abundance of thorium-90, which is at a lower concentration due to the fact that much of it has decayed away over the lifetime of the star. Figure from Cowan et al. 1997.
So, perhaps for every five atoms of europium is produced, theory might expect the production of one atom of thorium. This, then, offers a method of estimating the initial concentration: simply measure the concentration of a nonradioactive element (such as europium) and multiply by the constant given by theory determine the initial concentration of the radioactive element (thorium). Comparison of that to the currently observed value then gives us the age of the elements contained within the star. This is illustrated in Figure 5.

Note that this age is not exactly equal to the age of the star itself; it is essentially the "mean age" of the supernovae that formed it, which themselves may not have happened all at the same time either. For very old stars thought to have formed in the very first few million years after the universe this effect is not significant, but measurements of ages of younger stars (such as the Sun) will generate ages much older than that of the star itself.

The best constraints on the age of the universe are, of course, expected to be found on stars we believe to be oldest. As a result, most stars dated by this method are stars we expect from other observations (most importantly, the observation of very low abundances of heavy elements in general, since the Big Bang created almost no heavy elements and the oldest stars would be expected to have similarly low metal abundances).

AuthorsIsotopeStarAge
Cowan et al. (1997) 232Th CS 22892-052 15.2 +/- 3.7 Gyr
Cowan et al. (1999) 232Th HD115444 15.6 +/- 4.6 Gyr
Cayrel et al. (2001) 238U CS 31082-001 12.5 +/- 3 Gyr
Wanajo et al. (2002) 238U CS 31082-001 14.1 +/- 2.5 Gyr
 
Table 2 - Ages derived from various recent studies of radioactive elements in old stars.
Ages calculated by this method trend towards the high side of expectation: thorium-232 studies of two old stars by Cowan et al. (2002) report ages of about 15.2 and 15.6 billion years, though with large error estimates of 3.7 and 4.6 billion years on the two measurements these are not clearly inconsistent with current cosmological models. Uranium-238 lines have also recently been used to measure the ages of old stars and have generated ages that agree better with cosmological theory: 12.5 +/- 3 billion years for the star CS 31082-001 (Cayrel et al., 2001) and 14.1 +/- 2.5 billion years for the star CS 31082-001 (Wanajo et al., 2002). These results are summarized in Table 2.



Figure 6 - Graphical illustration of stellar evolution of a sunlike star. This process takes several billion years.
Stellar Evolution Methods

Main-sequence turnoff dating of star clusters

Stars do not have infinite lifetimes. Most stars generate their energy by fusing hydrogen into helium in the core, but with a limited supply of hydrogen actually available, this process can only continue for a limited time when the hydrogen supply is exhausted in the core, fusion progresses to other locations and eventually to heavier elements, causing the properties of the star to change dramatically. In general, once its core hydrogen is exhausted, a star swells in size and increases in luminosity to become a "red giant", and then destroys itself a relatively short time later either gently, by ejecting its outer layers to leave behind its denuded core (which itself cools to become a faint star known as a white dwarf), or violently, by exploding in a supernova. The former case is illustrated in Figure 6.

The time a star takes to expend its central hydrogen is well-known, easily calculable given the mass of hydrogen in the star's core and its luminosity: tlife = M c2 e / L, where e is the efficiency of hydrogen fusion (about 0.7%). Furthermore, the core mass is proportional to the total mass, while luminosity of a hydrogen-burning star is determined almost entirely by the mass itself by the approximate relation L ~ M4 (or more accurately L ~ M3.7) so the lifetime of a star is governed by the approximate proportionality tlife ~ M-2.7. Therefore, by knowing a hydrogen-burning star's mass, we can determine its lifetime, which imposes a maximum age.

The first challenge in using this as a dating method is separating these hydrogen-burning stars from the red giants (which are quite unstable and do not obey any of the relations above) and the white dwarfs (which do not produce energy and are effectively have an infinite lifetime). As mentioned previously, giant stars are very luminous but so are the most massive "normal" hydrogen-burning stars. They are also cool, and generally appear red in color as a result but so do the least massive normal stars.
Figure 7 - The H-R diagram. Normal stars like the Sun occupy the "main sequence" region. Stars which have exhausted their central hydrogen are much larger and are called "giants"; these eventually eject their outer envelope to become tiny white dwarfs.
However, a combination of these two factors is sufficient to identify which stars have exhausted their central hydrogen supply a star that is luminous and cool, conversely or a star that is dim but hot, cannot be an ordinary star. The H-R diagram, one of the most important plots in astronomy, represents this distinction graphically: for a group of stars, the temperature is plotted on the x-axis (increasing to the left) while the luminosity is plotted on the y-axis. When plotted on such a diagram, all hydrogen-burning stars lie along a diagonal band, with the hottest stars also being the most luminous; the coolest stars are the least luminous. This feature is referred to as the "main sequence". By contrast, red giants occupy the upper right corner of the diagram (high luminosity, low temperature) while white dwarfs occupy the opposite corner (low luminosity, high temperature). A schematic of the H-R diagram is presented in Figure 7.

Once a star is known to be a main-sequence star, its lifetime can be determined easily by the relations above: L ~ M^4 and t ~ M-3, so t ~ L-0.75. The Sun's main-sequence lifetime is about 10 billion years, so a star 10,000 times as luminous will live only 10 million years.

However, this is an estimate of the star's maximum age: the exact age for any star cannot be determined from this analysis, since main-sequence stars are generally very stable and older stars are very difficult to distinguish from younger stars (though there are some small differences that can be used in some cases). Fortunately, this problem is quite surmountable for indeed while a single star's age may be uncertain, the properties of a group of stars can be used to date an object precisely.

Imagine that a star population is found where the constituent stars all formed at the same time, but with a wide variety of masses. Initially these objects will begin hydrogen fusion, and all fall along the main sequence. Immediately, however, we can see that this situation will not last the most luminous stars are short-lived, and will soon disappear from the main sequence, while the less luminous stars will live for a longer (but still finite) length of time. So as time advances, the observed main sequence tends to shorten, like a burning fuse, as the stars expend their hydrogen and enjoy a brief stint as giants before ultimately permanently settle toward the bottom-left corner as white dwarfs. Therefore, the length of the main sequence or, more specifically, the luminosity and temperature of the "turnoff" point where stars depart from the main sequence to become giants indicates the age. A star at this turnoff point must be at the end of its lifetime (as all stars more massive than it have already expended their central hydrogen) and therefore its lifetime is its age. The evolution of the H-R diagram for such a stellar population is illustrated in Figure 8.
Figure 8 - An animated H-R diagram for population of stars of different masses, all of which were created simultaneously. As the population ages, the high-mass stars disappear from the main sequence first, followed by lower-mass stars (which have longer lifetimes). The turnoff point of the main sequence is therefore an indication of a the population's age.

Figure 9 - Images H-R diagrams of two clusters, the open cluster M67 (a young cluster), and the globular cluster M4 (an old cluster). The main sequence is significantly shorter for the older cluster; the luminosity and temperature of stars at the 'turnoff point' can be used to date these clusters.
Are such star populations found? Certainly - they are referred to as star clusters, and form when a large gas cloud collapses in a chaotic manner, fragmenting into a large number of smaller collapsing clouds of varying masses, eventually resulting in a population of stars that is not only the same age, but also the same position in the sky, the distance, and the same chemical composition eliminating numerous other sources of systematic error in the process! Such clusters come in two kinds open clusters, loose aggregations of a small number of stars found in the Galactic plane, and globular clusters, concentrated spherical clusters of thousands of stars each, which orbit the center of the Galaxy outside the plane.


Typical open and globular clusters are presented plotted in Figure 9. The difference in age is immediately apparent: the open cluster's main sequence is very long, while the globular cluster's is short, with a large population of evolved giant stars (white dwarfs are too faint to show up in either case). This difference is in fact typical open clusters tend to be quite young, since over time their stars drift apart and no longer appear as a cluster... making them not particularly useful for constraining the age of the universe. Globular clusters, however, are tightly bound and effectively permanent and furthermore,
Authors Age
Chaboyer et al. 1997 14.6 +/- 1.7 Gyr
Gratton et al. (1997) 11.8 +/- 2.3 Gyr
Reid et al. (1997) 12-13 Gyr
Chaboyer et al. (2001) 11.5 +/- 1.3 Gyr
 
Table 3 - Recent main-sequence turnoff measuremens of the ages of several globular clusters.
all appear to date from about the same epoch of 10-15 billion years ago. Though there were some early concerns when globular cluster dates were found to be marginally incompatible with the age of the universe, the elimination of an error in estimating the distance to these clusters (and, in turn, the luminosities of the constituent stars) means that current globular cluster dates are quite compatible with the current cosmological model: some recent dates are presented in Table 3. Even adding in the minimum time necessary for the globular cluster itself to form (about 500 million years) the most detailed recent studies appear to give ages in good agreement with cosmology.



White dwarf cooling times
Figure 10 - Schematic of a white dwarf star.
Stars that leave the main sequence do not entirely lose their relevance. While giant stars are extremely unstable objects, changing in luminosity by enormous factors over periods ranging from thousands of years to just hours, white dwarfs the stellar endstate of all but the most massive stars - are very simple, and very stable. White dwarfs form when a giant star ejects its outer envelope, leaving behind the core of "spent" nuclear fuel (usually carbon and oxygen with some helium; Figure 10). White dwarfs no longer undergo nuclear reactions, but initially retain the extremely high temperatures of the stellar core itself. Supported by quantum-mechanical degeneracy pressure, the white dwarfs then radiate this energy away slowly as electromagnetic radiation, and cool very gradually with time.

This slow cooling allows us another stellar dating method. In general, white dwarfs have approximately the same mass and all begin at approximately the same temperature; and with no nuclear or chemical reactions and no gravitational contraction occurring, their rate of cooling with time is extremely simple. So by measuring the temperature and luminosity of any white dwarf (that is to say, its position on an H-R diagram), we can estimate its approximate age since it formed from the death of its parent star. And while any given white dwarf may be relatively young and therefore not a useful constraint on the age of the universe, since white dwarfs are permanent objects, for a sufficiently large sample, in a large enough sample we should expect at least a few the oldest white dwarfs should originate from the earliest stars.

Figure 11 - H-R diagram of white dwarfs in a globular cluster: 'I' is a measure of luminosity, 'V-I' a measure of temperature (temp. increases to the left). The hottest white dwarfs are at the top left; these are from stars which left the main sequence only recently. The colder, fainter white dwarfs near the center of the plot have cooled for a much longer time; their temperature and luminosity has been used to date the age of this cluster. (Part of the main-sequence of the cluster is visible at the top right.) Taken from Fig. 1 of Hansen et al. 2002).
The best place for this kind of survey is again globular clusters - in addition to having a large population of stars expected to be old in the first place, the high concentration of stars makes it easier to image large numbers of these extremely faint objects in less time. An example is plotted in Figure 11: the population of white dwarfs is quite evident, with the older, cooler dwarfs closest to the bottom right and the younger, hotter dwarfs higher and to the left. This age difference is due to the varying ages at which the white dwarfs actually emerge from their progenitor stars: the hotter, "younger" dwarfs came from longer-lived stars (see Figure 8). The cooler, fainter white dwarfs, however, formed from much shorter-lived main-sequence stars, and have spent almost their entire lifetime in their current degenerate state. The shortest-lived stars which actually form white dwarfs have an initial mass of about 8 solar masses (any more and they explode as supernovae without forming a white dwarf); these stars lived as main-sequence stars for only about 50 million years and as giants for an even shorter time. (This short duration is nearly negligible compared to the cooling time as a white dwarf, but is nominally accounted for in estimating the total age of the star.)

The ages of white dwarf populations are generally in good agreement with cosmological models. The white dwarfs in the cluster plotted in Figure 11 have an age of about 12.7 +/- 0.7 billion years (Hansen et al. 2002); another cluster population dated to about 12.1 +/- 0.9 billion years (Hansen et al. 2004). As with the main-sequence turnoff method, we expect the universe itself to be older by about 500 million years. The population of white dwarfs in the Galactic disk has also been dated by this method; with a corresponding minimum age of about 9.5 +/- 1 billion years (Oswalt et al. 1996).


Conclusion

A summary of the most recent evidence of each category is presented in Table 2, with adjustments made to the ages of objects within the universe based on the amount of time it is believed to take for these objects to form. All results are compatible with the cosmological estimate, and when averaged together estimate an age for the universe of about 13.5 billion years quite close to the cosmological 13.7 billion for an accelerating universe containing dark energy, but still far older than the 9-11 billion expected from a matter-dominated universe. It would appear, then, that our best long-term models are in good agreement with the current cosmological paradigm, and strengthen the case for dark energy.

AuthorsObjectAge
Cosmological: Various   13.7 +/- 0.2 Gyr
Radiometric: Cowan et al. (1999) HD 115444CS 14.5 +/- 3.0 Gyr
 Wanajo et al. (2002) CS 31082-001 16 +/- 5 Gyr
Main-sequence turnoff: Gratton et al. (1999) Multiple GCs 12.3 +/- 2.5 Gyr
 Chaboyer et al. (2001) Multiple GCs 12.0 +/- 1.5 Gyr
White dwarf cooling: Hansen et al. (2004) M 4 12.8 +/- 1.1 Gyr
 
Table 4 - Summary of recent measurement of the age of the Universe.



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