The age of the universe

The big bang theory is a remarkably simple theory built on two pillars: the theory of general relativity and the assumption that the universe is isotropic and homogeneous on large scales. This theory has had a number of important successes: it can explain the observed expansion of the universe; the thermal microwave background radiation; the observed abundances of deuterium, helium, and lithium; and the rapid evolution seen in distant galaxies (see refs. 1 and 2 for recent reviews). Yet this successful model faces a potential crisis: the age of the oldest stars may exceed the predicted age of the universe. In the first half of the session, Michael Bolte of the University of California, Santa Cruz, described efforts to measure the age of the oldest stars. Because the oldest stars ought to be younger than the universe, this places a lower bound on t₀, the age of the universe. In the second half of the session, Wendy Freedman of the Carnegie Observatories discussed measurements of the Hubble constant, H₀, the rate of expansion of the universe. In the simplest and best explained version of the big bang theory—a flat, matter-dominated universe—the age of the universe is 2/3H₀. The final section of this paper is based on the panel discussion of the implications of their results. The Appendix describes the relationship between the age of the universe and the Hubble constant in different cosmologies and explains why many cosmologists believe that the universe is flat.

Determining the Ages of the Oldest Objects

Because the first generation of stars formed some time after the big bang, the age of the oldest known stars places a lower limit on the age of the universe.

Theory of Stellar Evolution.

Stars are remarkably simple systems: they are slowly evolving, nearly spherical clouds composed mostly of hydrogen and helium that can be accurately modeled on a computer. The basic physics needed to model the structure and evolution of stars is mostly well understood: nuclear cross-sections, the equation of state of matter, and the physics of hydrostatic equilibrium and radiation transfer. Although stellar structure does depend somewhat on the physics of convection, which remains poorly understood, stellar ages are relatively insensitive to the details of convection.

Main-sequence stars are stars, like our Sun, that fuse hydrogen to helium in their cores. For a given chemical composition and stellar age, a star’s luminosity, the total energy radiated by the star per unit time, depends only on its mass. Stars that are 10 times more massive than the Sun are more than 1,000 times more luminous than the Sun. We should not be too embarrassed by the Sun’s low luminosity: it is 10 times brighter than a star of half its mass. More massive main-sequence stars are also bluer (higher surface temperatures) than less massive stars. Thus, Sirius, which is more massive than the Sun, is blue and very luminous, whereas Alpha Centauri, which is less massive than the Sun, is red and less luminous.

The fundamental fuel for a star’s luminosity is mass. In any of the fusion reactions that result in hydrogen conversion to helium, a small amount of mass is transformed into energy in the form of neutrinos and γ-rays: the neutrinos flee the scene and the γ-rays are immediately absorbed, providing the heat source for the star. Because stars have only a limited supply of hydrogen in their cores, they have a limited lifetime, τ_MS, on the main sequence. This lifetime is proportional to ƒM/〈L〉, where ƒ is the fraction of the total mass of the star, M, available for nuclear burning in the core, and 〈L〉 is the time average luminosity of the star on the main sequence. Because of the strong dependence of luminosity on stellar mass, τ_MS ∝ M^−2.5, it is fortunate that our Sun is not more massive because high-mass stars rapidly exhaust their core hydrogen supply. Once a star exhausts its core hydrogen supply, the star becomes redder, larger, and more luminous, and it moves off the main sequence and becomes a red giant star.

Astronomers find it convenient to represent the properties of stars on a Hertzsprung–Russell (HR) diagram, a plot of a star’s luminosity and surface temperature. For historical reasons, optical astronomers like to plot the magnitude of a star, −2.5 times the base 10 logarithm of its luminosity, on the y axis, and the temperature of the star on the x axis. To further obscure the field, temperature increases to the left on the diagram.

The HR diagram is a particularly useful way to display the properties of stars in a cluster. A cluster is a dense collection of stars that are thought to have all formed at about the same time (give or take a million years). A very young cluster has main-sequence stars over a broad range of masses (and luminosities and temperatures). A 2 billion-year-old cluster contains main-sequence stars up to a “turn-off mass” of 2 solar masses—more massive stars exhaust their core hydrogen supply in under 2 billion years. A 10 billion-year-old cluster contains main-sequence stars up to a “turn-off mass” of 1 solar mass. A 15 billion-year-old cluster would contain no main-sequence star more massive than 0.85 solar mass and no main-sequence star more luminous than ≈0.5 times the luminosity of the Sun. Thus, by determining the maximum luminosity of a main-sequence star in a cluster, astronomers can measure its age.

Astronomers, however, cannot measure the luminosity of a star directly; they can only measure the flux from a star, F. If the distance to the cluster, D, can be determined, then energy conservation implies that L = 4πD²F. The challenge, and the major source of uncertainty in age determination, is measuring the distance to stellar clusters.

Age determinations also depend on the star’s chemical composition: a star’s evolution depends on its initial abundance of helium, carbon, oxygen, and iron, because these elements (and other less common elements) all affect the rate at which photons can escape from the core of a star and, particularly in the case of oxygen, moderate the energy generation reactions. Because the oldest stars have very low abundances of these elements, stellar age estimates for the globular clusters are, fortunately, not very sensitive to these details.

Should We Believe the Models?

Stellar models are needed to relate the observed luminosity and surface temperature of a star to its core mass and its time average luminosity, so that we can determine its main-sequence lifetime. Fortunately, the current stellar models are thought to be very reliable in the relevant mass range. The models are less reliable for very low-mass stars that contain complex molecules in their outer atmospheres and for higher-mass stars, whose cores are altered by convection.

The stellar models correctly predict the age and structure of the Sun. The age, total luminosity, surface temperature, and chemical composition of the Sun are accurately known and are well matched by the models (although, as the Sun is an important calibration point, this by itself is slightly suspect as a test of the models). The Sun also sustains an astonishingly large number of vibrational modes, and by studying the Sun’s oscillations, astronomers have measured the properties of the interior of the Sun, just as geologists have probed the Earth’s interior with seismology. The solar models agree remarkably well with the Sun’s observed properties (3).

The stellar models also correctly predict the temperature–luminosity relation for the nearby subdwarfs, main-sequence stars with chemical properties similar to stars in the oldest clusters. These stars are so close to the solar system that their distance can be determined by trigonometric parallax, so their luminosity can be accurately measured. There are now 11 dwarfs with accurate parallax distance: their luminosities and temperatures agree with model predictions (4).

Nature offers astronomers a wonderful laboratory for testing stellar models with double-line, eclipsing binary stars. Astronomers can measure the period of these binaries, their velocities, and the inclination of their orbits. With these measurements, it is possible to determine their masses to better than 1%. In probably their most stringent test, the observed mass, luminosities, and temperature are in excellent agreement with the stellar models. Future surveys should be able to detect more of these systems. Their detection and study in a globular cluster (see below) would be an important confirmation of the inferred ages.

Observing Old Stars in Old Clusters.

Globular clusters are thought to be the oldest clusters in the Galaxy. Globular clusters are dense spherical clusters of stars that are on orbits that suggest that they were formed in the initial collapse of our Galaxy. Their stellar densities are so high that the Hubble Space Telescope (HST) was needed to resolve their dense cores (see Fig. 1). Globular clusters are iron-poor: the relative abundance of iron to hydrogen in a globular cluster star is only 1/10 to 1/150 of the relative abundance in the Sun. These stars are also depleted in carbon, oxygen, and all of the other elements heavier than lithium. Because all of the elements heavier than lithium are synthesized in stars, these low abundances suggest that globular clusters formed very early in the history of the Galaxy before multiple generations of stars built up element abundances.

Figure 1.

Public archive HST image of M15, one of the nearby globular clusters. (Image taken by P. Guhathakurta. This figure was created with support to the Space Telescope Science Institute, operated by the Association of Universities for Research in Astronomy, Inc., from NASA contract NAS5-26555, and is reproduced with permission from AURA/STScI.)

Fig. 2 shows an HR diagram for M92, an extremely iron-poor globular cluster. The most iron-poor clusters also appear to be the oldest. The distance to the cluster was estimated by requiring that the main-sequence stars in the cluster have the same luminosity as nearby subdwarf stars with the same temperature (4). The lines on the plot are the model curves for 12, 14, 16, and 18 billion-year-old clusters with the appropriate chemical abundances. Fitting the data and propagating all of the observational errors implies a cluster age of 16 ± 2 billion years, a minimum age for the universe.

Figure 2.

HR diagram for M92. The squares are measured colors and brightnesses for individual stars in the cluster. The lines show model predictions for the positions of stars for cluster ages of 14, 16, and 18 billion years. The match of the models to the cluster data for an age of 16 billion years is remarkably good.

Measuring the Expansion Age of the Universe

Historical Overview.

In the 1920s, Edwin Hubble, using the newly constructed 100-inch telescope at Mount Wilson Observatory, detected variable stars in several nebulae, diffuse objects whose nature was a topic of heated debate in the astronomical community. His discovery was revolutionary, for these variable stars had a characteristic pattern resembling a class of stars called Cepheid variables. Earlier, Henrietta Leavitt, part of a group of female astronomers working at Harvard College Observatory, had shown a tight correlation between the period of a Cepheid variable star and its intrinsic luminosity. Thus, Hubble, by measuring the period of these stars and their fluxes, showed that these nebulae were not clouds within the Milky Way but were external galaxies far beyond the edge of our own Galaxy.

Hubble’s second revolutionary discovery was based on his plot (shown as Fig. 3) of galaxy distance determinations and measurements of the relative velocities of these galaxies. He showed that more distant galaxies were moving away from us more rapidly: the universe was not static, but rather was expanding. This discovery marked the beginning of the modern age of cosmology. Today, Cepheid variables remain the best method for measuring distances to galaxies and are vital to determining the expansion rate and age of the universe.

Figure 3.

This figure shows Hubble’s original diagram (5). Plotted on the y axis is v, the velocity of each galaxy in his sample relative to the Milky Way in km/s. Plotted on the x axis is the Hubble’s inferred distance to the galaxy, d, in parsecs (pc). Because of calibration problems, Hubble’s original distance estimates were off by nearly an order of magnitude. One parsec is approximately 3 light years or 3 × 10¹⁶ meters. “Hubble’s law” is a linear relationship between velocity and distance: v = H₀d, where H₀, the slope of the line in the diagram, has units of km/s/Mpc (where 1 Mpc = one million parsecs). Hubble’s constant, H₀, can be rewritten in more conventional units: 100 km/s/Mpc ≈ 1 × 10⁻¹⁰ yr⁻¹.

What Are Cepheid Variables?

The structure of all stars, including the Sun and Cepheid variable stars, is determined by the opacity of matter in the star. If the matter is very opaque, then it takes a long time for photons to diffuse out from the hot core of the star, and strong temperature and pressure gradients can develop in the star. If the matter is nearly transparent, then photons move easily through the star and erase any temperature gradient. Cepheid stars oscillate between two states; when the star is in its compact state, the helium in a layer of its atmosphere is singly ionized. Photons scatter off of the bound electron in the singly ionized helium atoms. Thus, the layer is opaque, and large temperature and pressure gradients build up across the layer. These large pressures cause the layer (and the whole atmosphere) to expand. When the star is in its expanded state, the helium in the layer is doubly ionized, so that the layer is more transparent to radiation and there is a much weaker pressure gradient across the layer. Without the pressure gradient to support the star against gravity, the layer (and the whole atmosphere) contracts, and the star returns to its compressed state.

Cepheid variable stars have masses between 5 and 20 solar masses. The more massive stars are more luminous and have more extended envelopes. Because their envelopes are more extended and the density in their envelopes is lower, their variability period, which is proportional to the inverse square root of the density in the layer, is longer.

Difficulties in Using Cepheids.

A number of difficulties have been associated with using Cepheids as distance indicators. Until recently, astronomers used photographic plates to measure the fluxes from stars. The plates had a highly nonlinear response and often produced inaccurate flux measurements. Because massive stars are short lived, they are always located near their dusty birthplaces. Dust absorbs light, particularly at blue wavelengths where most photographic images were taken, and if not properly corrected for, this dust absorption could lead to erroneous luminosity determinations. Finally, it has been difficult to detect Cepheids in distant galaxies from the ground because Earth’s fluctuating atmosphere makes it impossible to separate these stars from the diffuse light of their host galaxies.

Another historical difficulty with using Cepheids as distance indicators has been the problem of determining the distance to a sample of nearby Cepheids. In recent years, astronomers have developed several reliable and independent methods of determining the distances to the Large Magellanic Cloud (LMC), one of the satellite galaxies of our own Milky Way Galaxy. Because the LMC contains large numbers of Cepheids, it can be used to calibrate the distance scale.

Recent Progress.

Recent technological advances have enabled astronomers to overcome a number of the other past difficulties. New detectors called CCDs (charge-coupled devices) make accurate flux measurements possible. These new detectors are also sensitive in the infrared wavelengths. Dust is much more transparent at these wavelengths. By measuring fluxes at multiple wavelengths, astronomers are able to correct for the effects of dust and make more accurate distance determinations.

These advances enabled accurate study of the nearby galaxies that comprise the “Local Group.” Astronomers observed Cepheids in both the metal-rich inner region of M31 (Andromeda) and its metal-poor outer region (6). This work showed that the properties of Cepheids did not depend sensitively on chemical abundances. Despite these advances, astronomers, limited by the Earth’s atmosphere, could only measure the distances to the nearest galaxies. In addition to the motion due to the expansion of the universe, galaxies have “relative motions” due to the gravitational pull of their neighbors. Because of these peculiar motions, astronomers need to measure the distances to distant galaxies so that they can determine the Hubble constant.

To push deeper into the universe, astronomers have developed a number of new techniques for determining relative distances to galaxies. These independent relative distance scales now agree to better than 10% (7). For example, there is a very tight relation, called the Tully–Fisher relation, between the rotational velocity of a spiral galaxy and its luminosity (8). Astronomers also found that type Ia supernovae, which are thought to result from the explosive burning of a white dwarf, all had nearly the same peak luminosity (9). However, without accurate measurements of distance to large numbers of prototype galaxies, astronomers could not calibrate these relative distance measurements. Thus, they were unable to make accurate determinations of the Hubble constant.

Over the past few decades, leading astronomers, using different data sets, reported values for the Hubble constant that varied between 50 and 100 km/s/Mpc (5 × 10⁻¹¹ to 1 × 10⁻¹⁰ yr⁻¹) with groups claiming errors as small as 5 km/s/Mpc (9). Resolving this discrepancy, which corresponds to a factor 2 uncertainty, is one of the most important outstanding problems in observational cosmology.

Hubble Key Projects.

One of the key projects of the HST is to complete Edwin Hubble’s program of measuring distances to nearby galaxies. Although the HST is comparable in diameter to the Carnegie Institution’s telescope on Mount Wilson used by Hubble, it has the advantage of being above the Earth’s atmosphere, rather then being located on the outskirts of Los Angeles. Thus, HST can resolve Cepheids in more distant galaxies. The key projects aim to determine the distances to 20 nearby galaxies. With this large sample, HST can calibrate and cross-check a number of the secondary distance indicators. HST will also be able to check if the properties of Cepheid variables are sensitive to stellar composition.

NASA’s repair of the HST restored its vision and enabled the key project program. Fig. 4 shows several images of M100, one of the nearby galaxies observed by the key project program. Note that with the refurbished HST, it is much easier to detect individual bright stars in M100, a necessary step in studying Cepheid variables. The key project has now detected about 50 Cepheid variables in M100 and determined a distance of 16.1 ± 1.8 Mpc for the galaxy (10–12). Because M100 is close enough to us so that its peculiar motion (its motion induced by the gravitational influence of nearby mass concentrations) may be a significant fraction of its Hubble expansion velocity, the key project team used relative distance indicators to extrapolate from the Virgo cluster, a cluster containing M100, to the more distant Coma cluster and to obtain a measurement of the Hubble constant:

The dominant source of error is a result of the extended angular extent of the Virgo cluster on the sky and the uncertainty of knowing where M100 lies with respect to the center of the cluster. This error will be reduced when more distances to Virgo cluster galaxies are obtained.

Figure 4.

M100, one of the spiral galaxies used in the Hubble constant key project analysis, as seen by prerepair (Left) and postrepair HST (image from HST Public Archive STScI-Pr94-01). (This figure was created with support from the Space Telescope Science Institute, operated by the Association of Universities for Research in Astronomy, Inc., from NASA contract NAS5-26555, and is reproduced with permission from AURA/STScI.)

The key project determination of the Hubble constant is consistent with a number of independent efforts to estimate the Hubble constant. A recent statistical synthesis (13) of the published literature yields 66 < H₀ < 82 km/s/Mpc as a 95% confidence interval. However, there still is not a complete consensus on the value of the Hubble constant; a recent analysis using type Ia supernovae yields a value for the Hubble constant that is formally inconsistent with many previous measurements: H₀ = 47 ± 5 km/s/Mpc (14).

In the past year, the key project has detected Cepheids in eight other galaxies, and the results are consistent with those from M100. These new observations make possible a number of important checks and calibrations. In M101, the key project has detected Cepheids in both metal-poor and metal-rich regions; this will enable a test to see if the Cepheid properties depend on abundances. A particularly important measurement is the determination of the distance to the Fornax cluster, a nearby group of galaxies used to calibrate the supernova, in addition to three other relative distance scales. This measurement will be important for understanding the remaining discrepancies. Ultimately, the key project should be able to make a reliable measurement of the Hubble constant that is accurate to better than 10%.

Discussion and Implications‖

The lower limit on the age of the universe can be combined with the key project determination of the Hubble constant to yield a dimensionless number,

Here, we optimistically set the age of the universe equal to the age of the oldest stars, clearly an underestimate. If the universe is flat and matter-dominated, then theory predicts H₀t₀ = 2/3. This discrepancy does not yet have enormous statistical significance and may reflect some unidentified set of problems in either the theory of stellar evolution, the cluster distance scale, or extragalactic distance scales. However, it also may be the signature of missing physics in the big bang theory.

The determination of the age of the universe from the Hubble constant depends on the total density and on the composition of matter in the universe. The prediction that H₀t₀ = 2/3 assumes that the universe was composed mostly of “normal” nonrelativistic matter and that the density of the universe was sufficient to make its geometry flat (see Appendix for further discussion). By relaxing either of these assumptions, we alter the prediction for H₀t₀.

Cosmological Constant.

Einstein first proposed the cosmological constant, Λ, as a mathematical fix to the theory of general relativity. In its simplest form, the theory predicted that the universe must either expand or contract. Einstein thought the universe was static, so he added this new term to stop the expansion. Friedman, a Russian mathematician, realized that this was an unstable fix and proposed an expanding universe model. When Hubble’s study of nearby galaxies showed that the universe was expanding, Einstein regretted modifying his elegant theory and viewed the Λ term as his “greatest mistake.”

Many cosmologists advocate reviving the Λ term. Modern field theory associates this term with the energy density of the vacuum.** For this term to be cosmologically interesting, its value would need to be ∼ (10⁻⁴ eV)⁴ and would either require new physics on the milli-eV scale or some minute (10⁻¹²⁰) correction to quantum gravity; either way, the addition of a cosmological constant term has profound implications for particle theory (15).

The advantage of the cosmological constant term is that it significantly improves the agreement between theory and observation (16, 17). If the cosmological constant today contributes most of the energy density of the universe, then the extrapolated age is much larger. Adding a cosmological constant term to the inflationary model leads to a model that appears to be consistent with the observed large-scale distribution of galaxies and clusters (18), measurements of cosmic background fluctuations (19), and properties of x-ray clusters (20).

The cosmological constant term alters the relationship between distance and time. Specifically, for a positive value for Λ/3H₀², the volume of the universe and look-back time at a given redshift increases. One of the observational consequences of this altered geometry is a much higher probability for “multiple image gravitational lenses.” Multiple image gravitational lenses can arise when light traveling from a distant quasar toward our telescopes passes through a galaxy or a cluster of galaxies. The mass along the line of sight serves as a lens and can produce multiple images of the distant quasar. Turner (21) showed that the probability of multiple image gravitational lenses was much higher in a universe with a large cosmological constant term. Thus, if Λ was large, the HST would show multiple images nearly every time it was used to observe a distant quasar. This was not seen in a “snapshot survey” (22) or in large ground-based surveys (23). Based on this lack of lenses, Chris Kochanek, one of the panelists, has placed a 95% confidence level upper limit of Ω_Λ < 0.66, which eliminates much of the “allowed parameter” space (24).

If nonrelativistic matter makes up only 25% of the total mass of the universe, then the remaining “stuff” could be in some new form. The cosmological constant is only one possible way of modifying the equation of state of the universe. Other possibilities include non-Abelian gauged strings or a new form of matter. These exotic suggestions all have distinctive observational signatures that are testable in the next 5–10 years.

Is the Universe Open?

Another possible solution to the age problem is abandoning the assumption that the universe is flat. There is little observational evidence that the universe is flat, only theoretical prejudice.

If the universe is open and matter-dominated, then the predicted age for fixed Hubble constant is larger than a flat matter-dominated universe, but smaller than a flat universe with a cosmological constant and the same matter content. The Hubble constant predicts H₀t₀ ∼ 0.8 for cosmologically interesting parameters. An open matter-dominated inflationary model, like its cosmological constant-dominated cousin, appears to be consistent with the observed large-scale distribution of galaxies and clusters, measurements of cosmic background fluctuations (25, 26), and properties of x-ray clusters (20). However, unlike the cosmological constant model, the open matter-dominated model does not predict excessive numbers of gravitational lenses. On the other hand, the range of Hubble’s constant and age consistent with the model is smaller (25); the model requires H₀ ∼ 60–65 km/s/Mpc and t₀ < 12 Gyr, marginally consistent with the observations discussed earlier.

Future Observations.

Observations in the next few years will clarify the age problem and, perhaps, point to its resolution. Planned observations of nearby galaxies will cross-check the distance scale, reduce concerns about systematic errors, and shrink the error bars on H₀. Similarly, future observations of binary systems and of nearby subdwarfs will further test stellar evolution models and increase confidence in determinations of t₀.

Observations of the microwave background will provide an independent probe of these basic parameters. Planned all-sky high-resolution maps of the microwave background will test the inflationary model and make independent measurements of the Hubble constant, Λ and Ω, potentially accurate to better than 5% (27). For example (28), if the universe has a cosmological constant, the dominant scale for microwave background fluctuations will be 1°, if it is open, the dominant scale will be 0.5°. Observations, not our theoretical prejudices, will resolve this issue.

ABBREVIATIONS

HR

Hertzsprung–Russell

HST

Hubble Space Telescope

Age of the Universe in the Big Bang Theory

If the universe is uniform, then general relativity has a simple solution for the evolution of the geometry of the universe: it either expands (or contracts) uniformly. The theory implies that two distant galaxies that today are separated by some distance x will at some later time be separated by a distance r = a(t)x. This expansion is consistent with the observed relation between a galaxy recessional velocity and its distance:

1

where dot denotes derivative with respect to time.

The density and composition of the universe governs its rate of expansion. The larger the density of the universe, the more effective gravity is at slowing its expansion. The evolution equation for a, derived rigorously from an exact solution of general relativity,

2

can also be computed from Newtonian theory. The composition of the universe determines its equation of state, p(ρ), where p is the pressure and ρ is the density. If the universe is composed mostly of ordinary (cold) matter, p = 0. If a cosmological constant dominates the density of the universe, then p = −ρ, and ρ is a constant. Cosmologists can estimate the age of the universe by integrating Eq. 2 back to a = 0.

The density of the universe also determines its geometry. Cosmologists define a critical density, ρ_crit = 8πG/3H₀² and a parameter Ω, the ratio of the energy density of the universe to the critical density. If Ω is greater than 1, space-time is positively curved and the self-gravity of the universe is strong enough to eventually reverse the expansion of the universe, and the universe will eventually recollapse. If Ω is less than 1, then the universe will expand forever.

Theoretical cosmologists have strong prejudice that Ω should be very close to 1. Originally, this prejudice was based on the observation that Ω = 1 corresponds to the universe having zero total energy. This belief was reinforced by the theory of inflation, developed by A. Guth, A. Linde, and P. Steinhardt. The inflationary theory solves a number of the paradoxes inherent in the big bang theory. It explains why the universe is homogeneous on large scales, why the scale of the universe is much larger than the other fundamental scales in physics, and it can explain the large entropy of the universe. As a bonus, the theory also provides a mechanism for generating density fluctuations that can grow to form galaxies.

If Ω = 1 and the universe is composed primarily of matter, then the big bang theory makes a definite prediction about the product of the expansion rate of the universe, H₀, and the age of the universe, t₀:

This prediction can be altered if most of the mass density in the universe is in some new form.