This implies that the initial density fluctuations in the Universe were also very small.
The current energy density of the microwave background is quite small, as might be expected for the thermal radiation from something that is colder than liquid helium. The number density of microwave photons is 410 cm−3, and the average energy per photon is 0.00063 electron volt (eV). Thus, the energy density is only 0.26 eV/cm3. This is 20,000 times less than the critical density. But when the Universe was very young (e.g., t = 0.5 × 10−6 t0, about 7,000 years after the Big Bang) and the scale factor was very small, a(t) = 10−4, the number density of photons was much greater, 4.1 × 1014 cm−3, and the energy per photon was also much greater, 6.3 eV. The photon density and average energy per photon correspond to a hotter blackbody with a temperature T = T0/a(t) = 27,250 K. As the Universe expands, it also cools. Thus, the energy density of the background when the Universe was small, dense, and hot was very large, 2.6 × 1015 eV/cm3 when a(t) = 10−4, and dominated the density of the Universe for all times less than 50,000 years after the Big Bang.
The very small temperature fluctuations indicate corresponding density fluctuations of about 33 parts per million 10,000 years after the Big Bang. Once the energy density of background radiation becomes less than the density of matter, the fluctuations grow as the denser regions gravitationally attract more material. The process of gravitational collapse makes the fluctuations grow in proportion to the scale factor a(t). Thus, in the case of a Universe with the critical density described before, the Universe was no longer radiation-dominated when a = 0.0003, so the fluctuations grew from 33 parts per million to 11%. This is just enough to explain the observed clustering of galaxies that we see in the Universe now. But if the Universe had no dark matter, then a = 10−3 when the Universe stopped being radiation-dominated. Furthermore, ordinary matter interacts with light and could not move through the background radiation until the Universe was cold enough for neutral hydrogen atoms to be stable. This happened approximately 400,000 years after the Big Bang, when the temperature of the microwave background fell to 3000 K, which, coincidentally, is when a = 10−3. Thus, if only ordinary matter had been present, the fluctuations implied by the COBE observations would have grown only to 3.3% at the current epoch. Such small density contrasts would be completely inconsistent with the far higher density contrasts we currently observe on many scales, such as galaxy clusters with amplitudes of well over 100%.
Light Element Abundances
Although the energy density of the microwave background dominated the Universe for the first 50,000 years after the Big Bang, it is even more significant during the first 3 minutes. One second after the Big Bang, the average energy of a photon was 3 million electron volts (MeV), which is a gamma ray. Gamma rays destroy any atomic nucleus; thus, 1 second after the Big Bang there were only protons (p or hydrogen nuclei), free neutrons (n), electrons (e), and positrons (e+), neutrinos (ve− νμ and ντ), and their antiparticle counterparts, the antineutrinos. The three types of neutrinos correspond to the three “families” of elementary particles, but, except for the neutrinos, all the particles in the second and third families (such as muons and tauons), are so heavy and unstable that they decay during the first second after the Big Bang. Weak nuclear interactions such as
determined the ratio of neutrons to protons. The neutron is heavier than the proton; the neutron-to-proton ratio declines as the temperature falls. But eventually, at about 1 second after the Big Bang, the density of electrons and neutrinos falls to such a low level that reaction (9) is no longer effective. After this time, the neutron-to-proton ratio gradually falls because of the radioactive decay of the neutron,
which has a half-life of 615 seconds.
As neutrons decay, the Universe expands and grows colder. Eventually the temperature falls to the point where the simplest nucleus, the heavy hydrogen or deuterium nucleus (d, the nucleus having both one proton and one neutron), is stable. This occurs when the temperature is about 109K, which occurs about 100 seconds after the Big Bang. At this point, the reaction
very quickly converts all neutrons into deuterium nuclei. Once deuterium is formed, it is quickly converted into helium through a network of interactions, with the net effect
Because almost all neutrons that survive until T is less than 109K end up bound in helium nuclei, the helium abundance in the Universe provides a measurement of the time it takes for the Universe to cool to 109K. If the Universe cools rapidly, there is a large helium abundance, but slow cooling gives low helium abundance because more of the neutrons decay. The standard Big Bang model, with three types of neutrinos, predicts a helium abundance that is correct to within the 1% margin of uncertainty of current observations. Reaction (12) requires collisions between two nuclei, and if the density of atomic nuclei is low, then a fraction of the deuterium will not react. Thus, the residual fraction of deuterium in the Universe is a sensitive measure of the density of atomic nuclei. Based on the abundance of deuterium and other light isotopes like 3He, the best estimate for the current density of nuclei of all sorts is equivalent to 1/4 hydrogen atoms per cubic meter (Copi, Schramm and Turner, 1995, Schramm, 1995). This is about 25 times less than the critical density. Because the density of the Universe must be close to the critical density to produce the observed clustering of galaxies, we find from the light element abundances that most of the mass of the Universe must be the mysterious dark matter.
In the 1940s, George Gamow and colleagues proposed that all the chemical elements were produced in the Big Bang. This proposal, described further in Virginia Trimble's Chapter 3, led to a prediction of a 5-K microwave background (Alpher and Herman, 1948), but this prediction was not followed up. The eventual discovery of the microwave background in 1964 was accidental. Why was this prediction ignored? The absence of stable nuclei with atomic weights of 5 and 8 means that the Big Bang produces only hydrogen and helium isotopes and a very small amount of lithium. When a model is supposed to produce all the elements from Z = 1 to 92, but actually only works for Z = 1, 2, and 3, its other predictions tend to be ignored. But in this case the predictions were right.
Horizons
We can see only a finite piece of the Universe. The naive estimate for how far we can see is ct0, the speed of light times the age of the Universe. This is, in fact, the distance traveled by photons coming from the most distant visible parts of the Universe, as measured by the photons. But when one defines distances in an expanding universe, the convention is to measure all intervals at the current time, t0. Because the Universe has expanded since t = 0, the earlier parts of the photon’s journey get extra credit. We can compute the distance we can see in a critical density universe by dividing the age of the Universe into more and more intervals. With one interval, we get ct0. With two intervals, we get 0.5 ct0/0.52/3 + 0.5 ct0 = 1.29 ct0 because the first half of the journey has expanded by the factor 1/a(t0/2) = 1/0.52/3. With four intervals, we obtain 1.58 ct0. With a very large number of intervals, we get 3 ct0 which is the distance to the horizon. For t0 ~ 13 Gyr, this is 40 billion light years.
Consider now an observer