The Age Of The Universe
As we noted in the cosmology lecture the observed increase in redshift Z with distance is interpreted, by most scientists, as evidence of the expansion of the universe. This led to the idea that the universe may have gone through a dense hot phase, with the whimsical name of the big bang, a finite time ago. The hot dense phase is generally regarded as the beginning of the universe, and the time since the beginning is, by definition, the age of the universe. A basic, and natural, question to ask is: How old is the universe?
It is important to realize that this question is undefined until one specifies 1) a definition of what one means by the age of the universe and 2) how it should be measured; that is, one needs a theory for the age of the universe. This was recognized as long ago as 1658, when Bishop James Ussher determined the age of the universe thus: he defined the age of the universe as the time since creation (by God) and measured the age by adding the periods specified in the Bible, suitably corrected using available astronomical and historical records. He determined that the universe was created in 4004 BC, at a time that would have been sunset in Jerusalem, October 22.
Today, however, there is overwhelming evidence that the universe is considerably more ancient that previously thought. But, like Bishop Ussher we need a theory of the age of the universe. At present the best theory we have is Einstein's general relativity. When applied to the universe, assuming the Cosmological Principle (the universe is uniform and isotropic on very large scales), the theory provides a relationship between the mass density and the age of the universe.
Acccording to general relativity, the universe will either expand forever or recollapse after a finite time, depending on the value of its mass density, r. If the density is too great, that is, greater than some critical density, rc, the universe will recollapse; that is, it will be closed. If the density is below, or equal to, the critical density it will expand forever; the universe will be open. The critical density is given by
(1) rc = 3H02/8pG
= 8 * 10-27 (H0/65) kg/m3
= just over 4 hydrogen atoms per cubic meter,
where H0 is the Hubble constant and G is Newton's gravitational constant. We have assumed a Hubble constant of H0 = 65 km/sec/Mpc. It is customary to discuss the density of the universe in terms of the parameter W, defined by
(2) W = r/rc
In the Einstein-de Sitter model the universe is assumed to have a flat spatial geometry. For this case the age of the universe is predicted to be
(3) t0 = (2/3)/H0
Current models of the universe assume that W is close to one. For example, in the inflationary model a truly stupendous expansion is assumed to have occurred early in the history of the universe. About 10-34 seconds after the creation, when the temperature was perhaps one thousand trillion trillion degrees, a bubble no more than 10-24 cm might have undergone an expansion by a factor of maybe 1050! At about 10-32 seconds this hyper-expansion stopped, leaving the bubble 1026cm across (that is, 100 million light years). However, the bit of the universe we can observe would have been, at that early time, only a few centimeters across! After 10-32 seconds the normal big bang evolution takes over.
This may sound utterly crazy, but it explains a lot. The fantastic stretching of space renders the observable universe geometrically flat (with W=1). This is because the observable universe would be but a tiny patch of a universe that may be a trillion trillion times larger. Even if the geometry of the universe were highly curved the observable part of it is such a small patch that it would nonetheless appear to us as spatially flat. Moreover, any weird particles or spacetime distortions that may have been created then would have been diluted practically out of existence; so we would not expect to find any such oddities in our neck of the woods. Lastly, the microwave background radiation, which was already uniform and isotropic within the micro-bubble, would have remained so after the hyper-expansion. The latter explains why the radiation at opposite sides of the sky have the same temperature; these patches of sky were once much less than 10-24 cm apart.
Alas, W is observed to be about 0.25! Therefore, if we nonetheless insist that W = 1, we are forced to conclude that the luminous matter we observe forms only a part of the total energy in the universe. The deficit must therefore be made up by some other unknown source of energy. This could be dark matter. Much more interesting and much more mysterious: the possible existence of vacuum energy; or perhaps an admixture of the two.
When we use Eq. (3) to estimate the age of the universe we get an answer of about 10 billion years. Unfortunately, this is less than the age of the oldest stars found in globular clusters, like M13. It is thought that these stars are about 15 billion years old. Presumably, however, the universe cannot be younger than its contents. This problem has been the source of much controversy and debate recently. Some scientists have suggested that the big bang idea is wrong and should be abandoned. However, it could be that the ages of the oldest stars are too high, or that the assumptions of the inflationary model are wrong, or the measurement of Hubble's constant is in error. Or perhaps there is a missing ingredient. Or some combination of all of these.
Within the context of a general relativistic description of the the universe there are two critical parameters that determine the global properties of the universe: The Hubble constant H0 and the deceleration parameter q0. If the universe is expanding today, it must have been expanding faster in the past. This is because gravity has been slowing down the expansion. (Of course, if an anti-gravitational force exists it is conceivable that the expansion is actually accelerating.) The deceleration parameter, as the name suggests, is a measure of how much the expansion has slowed since the big bang.
We have already encountered Hubble's law
(4) Z = d H0/c,
where Z is the redshift of a galaxy and d its (proper) distance from us. This law is valid only for small redshifts, say less than about 0.2; that is, for galaxies that are not too distant. For very distant galaxies we need a more accurate formula. We need this because when we look at very distant galaxies we are looking into the distant past when the expansion rate was, presumably, greater. A more accurate formula that takes account of the universal deceleration is
(5) Z = d H0/c + (1+q0)(d H0/c)2/2.
In practice, cosmologists do not use the proper distance d, directly, but rather use a distance scale, called the luminosity distance, dL, which is determined by requiring that the inverse square law
(6) f = L/4pdL2
hold true even for an expanding universe. (In a static, flat universe both distance scales coincide.) Here f is the measured flux (energy per square meter per second) and L is the absolute luminosity (energy per second). The luminosity distance is convenient because it can be determined from measuring f and determining L, only.
If we measure d (or dL) and Z for many galaxies we should, in principle, be able to determine the two constants H0 and q0. The chief difficulty is measuring the distances. The basic technique is to use objects of known intrinsic luminosity L (standard candles) and use the inverse square law along with the apparent luminosity to infer the distance. But to measure distances of billions of light years one needs extraordinarily bright objects. Amongst the brightest objects in the universe are exploding stars, called supernovas.
In recent years there has been growing evidence that Type Ia supernovas are standard candles. Type Ia supernovas, thought to be exploding carbon-oxygen white dwarf stars, can be as bright as an entire galaxy, therefore, they can be seen over great distances. The white dwarf star forms a binary with a larger star from which it draws matter. When the total mass of the white dwarf and its accreted material exceeds the Chandresekhar limit of 1.4 solar masses the white dwarf star detonates, reaching a temperature in excess of a billion degrees.
The hypothesis that all Type Ia supernovae have the same intrinsic brightness is suggested by the idea that the stars explode when they reach the same critical mass. In practice, there is some variation in luminosity, as well as in the shape of the light curves. But by calibrating the light curves, that is, by checking the luminosity-distance relationship using nearer supernovas it is possible to take into account some of this variation. The distances to the nearer Type Ia supernovas is measured using Cepheid variables that reside in the same galaxy as the supernovas.
There is now ample evidence that W, due to the observable energy, is rather less than one. One way out of this difficulty, assuming one wants a unit value for W, is to assume that the energy density of the universe consists of ordinary observable energy plus the energy of, say, the vacuum; that is, we assume that W = WM + WL, where WL is the contribution from the vacuum, and WM is that from ordinary matter and energy. (The parameter L was introduced by Einstein in his (not fully successful) attempt to predict a static universe. Later he removed L from his equations, calling its prior introduction the greatest mistake of his life.)
But how can mere vacuum have energy? Well, according to modern physics the vacuum is a cauldron of virtual particles that come in and out of existence on very short timescsales. In fact, when an honest calculation is done of the energy of the vacuum comes out to a gigantic value; alas, the answer is wrong by about a factor of 10100! This is surely the worst prediction in the history of physics. It is a complete mystery why the energy of the vacuum is so much smaller than the prediction.
Nonetheless, if vacuum energy exists it can have a marked effect on the evolution of the universe. Many groups have tried to measure the W parameters using Type Ia supernovas. Here is a plot from a paper by S. Perlmutter et al., which shows how the distance (measured in terms of the magnitude---if you like, the faintness---mB) is related to the redshift.
The three black (solid) curves are predictions for a curved universe assuming (WM, WL) = (0, 0), top curve, (1, 0), middle curve and (2, 0), bottom curve. The dashed curves are calculated for a flat universe. From the top, the dashed curves correspond to (WM, WL) = (0, 1), (0.5, 0.5), (1, 0) and (1.5, -0.5). So, if the universe has a globally flat geometry it would appear that it contains some amount of vacuum energy. Careful fits by the Supernova Cosmology Project, of all the available data, assuming a flat universe, gives a value of WM = 0.25 +/- 0.06 (statistical) +/- 0.4 (systematic). The alternative model is a curved (open) universe. If this model is true then the data seems to suggest that the universe is empty, which it clearly is not! Stay tuned.