The Einstein–de Sitter universe
News •
In 1932 Einstein and de Sitter proposed that the cosmological constant should be set equal to zero, and they derived a homogeneous and isotropic model that provides the separating case between the closed and open Friedmann models; i.e., Einstein and de Sitter assumed that the spatial curvature of the universe is neither positive nor negative but rather zero. The spatial geometry of the Einstein–de Sitter universe is Euclidean (infinite total volume), but space-time is not globally flat (i.e., not exactly the space-time of special relativity). Time again commences with a big bang and the galaxies recede forever, but the recession rate (Hubble’s “constant”) asymptotically coasts to zero as time advances to infinity. Because the geometry of space and the gross evolutionary properties are uniquely defined in the Einstein–de Sitter model, many people with a philosophical bent long considered it the most fitting candidate to describe the actual universe.
Bound and unbound universes and the closure density
The different separation behaviours of galaxies at large timescales in the Friedmann closed and open models and the Einstein–de Sitter model allow a different classification scheme than one based on the global structure of space-time. The alternative way of looking at things is in terms of gravitationally bound and unbound systems: closed models where galaxies initially separate but later come back together again represent bound universes; open models where galaxies continue to separate forever represent unbound universes; the Einstein–de Sitter model where galaxies separate forever but slow to a halt at infinite time represents the critical case.
The advantage of this alternative view is that it focuses attention on local quantities where it is possible to think in the simpler terms of Newtonian physics—attractive forces, for example. In this picture it is intuitively clear that the feature that should distinguish whether or not gravity is capable of bringing a given expansion rate to a halt depends on the amount of mass (per unit volume) present. This is indeed the case; the Newtonian and relativistic formalisms give the same criterion for the critical, or closure, density (in mass equivalent of matter and radiation) that separates closed or bound universes from open or unbound ones. If Hubble’s constant at the present epoch is denoted as H0, then the closure density (corresponding to an Einstein–de Sitter model) equals 3H02/8πG, where G is the universal gravitational constant in both Newton’s and Einstein’s theories of gravity. The numerical value of Hubble’s constant H0 is 22 kilometres per second per million light-years; the closure density then equals 10−29 gram per cubic centimetre, the equivalent of about six hydrogen atoms on average per cubic metre of cosmic space. If the actual cosmic average is greater than this value, the universe is bound (closed) and, though currently expanding, will end in a crush of unimaginable proportion. If it is less, the universe is unbound (open) and will expand forever. The result is intuitively plausible since the smaller the mass density, the smaller the role for gravitation, so the more the universe will approach free expansion (assuming that the cosmological constant is zero).
The mass in galaxies observed directly, when averaged over cosmological distances, is estimated to be only a few percent of the amount required to close the universe. The amount contained in the radiation field (most of which is in the cosmic microwave background) contributes negligibly to the total at present. If this were all, the universe would be open and unbound. However, the dark matter that has been deduced from various dynamic arguments is about 23 percent of the universe, and dark energy supplies the remaining amount, bringing the total average mass density up to 100 percent of the closure density.
The hot big bang
Given the measured radiation temperature of 2.735 kelvins (K), the energy density of the cosmic microwave background can be shown to be about 1,000 times smaller than the average rest-energy density of ordinary matter in the universe. Thus, the current universe is matter-dominated. If one goes back in time to redshift z, the average number densities of particles and photons were both bigger by the same factor (1 + z)3 because the universe was more compressed by this factor, and the ratio of these two numbers would have maintained its current value of about one hydrogen nucleus, or proton, for every 109 photons. The wavelength of each photon, however, was shorter by the factor 1 + z in the past than it is now; therefore, the energy density of radiation increases faster by one factor of 1 + z than the rest-energy density of matter. Thus, the radiation energy density becomes comparable to the energy density of ordinary matter at a redshift of about 1,000. At redshifts larger than 10,000, radiation would have dominated even over the dark matter of the universe. Between these two values radiation would have decoupled from matter when hydrogen recombined. It is not possible to use photons to observe redshifts larger than about 1,090, because the cosmic plasma at temperatures above 4,000 K is essentially opaque before recombination. One can think of the spherical surface as an inverted “photosphere” of the observable universe. This spherical surface of last scattering probably has slight ripples in it that account for the slight anisotropies observed in the cosmic microwave background today. In any case, the earliest stages of the universe’s history—for example, when temperatures were 109 K and higher—cannot be examined by light received through any telescope. Clues must be sought by comparing the matter content with theoretical calculations.
For this purpose, fortunately, the cosmological evolution of model universes is especially simple and amenable to computation at redshifts much larger than 10,000 (or temperatures substantially above 30,000 K) because the physical properties of the dominant component, photons, then are completely known. In a radiation-dominated early universe, for example, the radiation temperature T is very precisely known as a function of the age of the universe, the time t after the big bang.
Primordial nucleosynthesis
According to the considerations outlined above, at a time t less than 10-4 seconds, the creation of matter-antimatter pairs would have been in thermodynamic equilibrium with the ambient radiation field at a temperature T of about 1012 K. Nevertheless, there was a slight excess of matter particles (e.g., protons) compared to antimatter particles (e.g., antiprotons) of roughly a few parts in 109. This is known because, as the universe aged and expanded, the radiation temperature would have dropped and each antiproton and each antineutron would have annihilated with a proton and a neutron to yield two gamma rays; and later each antielectron would have done the same with an electron to give two more gamma rays. After annihilation, however, the ratio of the number of remaining protons to photons would be conserved in the subsequent expansion to the present day. Since that ratio is known to be one part in 109, it is easy to work out that the original matter-antimatter asymmetry must have been a few parts per 109.
In any case, after proton-antiproton and neutron-antineutron annihilation but before electron-antielectron annihilation, it is possible to calculate that for every excess neutron there were about five excess protons in thermodynamic equilibrium with one another through neutrino and antineutrino interactions at a temperature of about 1010 K. When the universe reached an age of a few seconds, the temperature would have dropped significantly below 1010 K, and electron-antielectron annihilation would have occurred, liberating the neutrinos and antineutrinos to stream freely through the universe. With no neutrino-antineutrino reactions to replenish their supply, the neutrons would have started to decay with a half-life of 10.6 minutes to protons and electrons (and antineutrinos). However, at an age of 1.5 minutes, well before neutron decay went to completion, the temperature would have dropped to 109 K, low enough to allow neutrons to be captured by protons to form a nucleus of heavy hydrogen, or deuterium. (Before that time, the reaction could still have taken place, but the deuterium nucleus would immediately have broken up under the prevailing high temperatures.) Once deuterium had formed, a very fast chain of reactions set in, quickly assembling most of the neutrons and deuterium nuclei with protons to yield helium nuclei. If the decay of neutrons is ignored, an original mix of 10 protons and two neutrons (one neutron for every five protons) would have assembled into one helium nucleus (two protons plus two neutrons), leaving more than eight protons (eight hydrogen nuclei). This amounts to a helium-mass fraction of 4/12 = 1/3—i.e., 33 percent. A more sophisticated calculation that takes into account the concurrent decay of neutrons and other complications yields a helium-mass fraction in the neighbourhood of 25 percent and a hydrogen-mass fraction of 75 percent, which are close to the deduced primordial values from astronomical observations. This agreement provides one of the primary successes of hot big bang theory.
The deuterium abundance
Not all of the deuterium formed by the capture of neutrons by protons would be further reacted to produce helium. A small residual can be expected to remain, the exact fraction depending sensitively on the density of ordinary matter existing in the universe when the universe was a few minutes old. The problem can be turned around: given measured values of the deuterium abundance (corrected for various effects), what density of ordinary matter needs to be present at a temperature of 109 K so that the nuclear reaction calculations will reproduce the measured deuterium abundance? The answer is known, and this density of ordinary matter can be expanded by simple scaling relations from a radiation temperature of 109 K to one of 2.735 K. This yields a predicted present density of ordinary matter and can be compared with the density inferred to exist in galaxies when averaged over large regions. The two numbers are within a factor of a few of each other. In other words, the deuterium calculation implies much of the ordinary matter in the universe has already been seen in observable galaxies. Ordinary matter cannot be the hidden mass of the universe.
