Lecture 27: The Big Bang, GUTs, Inflation
The Hot Big Bang
Many predictions arise from the model of the Universe starting in a Hot Big Bang. Consider the 3° K radiation itself. Since matter and energy are equivalent from E=mc2, does the density contributed by the 3° K radiation contribute significantly to the total density in the Universe?
The equivalent matter density is then
So the density of radiation is ~1/1000th that of matter, far less than the critical density.
However, the radiation density has not always been so much less than the density of matter. In what follows, z is the redshift of the time under consideration and R is the size of the Universe.
is currently 1/1000th the matter density.
Where did the Big Bang occur?
=> Very important to realize that the "explosion" was a rapid increase in the size of the Universe itself. Big Bang did not have a center nor could an observe stand outside and watch it.
Conditions at t=0 in the Big Bang
How can we estimate the temperature at early times?
At the current time
In terms of energy density this becomes
Since there are ~6x10-28 kg/m3 of matter, each kg of matter has ~1014 Joules of photon energy surrounding it! If the kg of matter absorbed all of this energy, it would have T~4x109° K!
Another line of argument comes from considering the relative numbers of photons and nucleons (meaning protons and neutrons) in the Universe today:
We can compute the number of photons per m3 from the CBR:
Compare this with the number of nucleons:
So the number of CBR photons per m3 is 109 times the number of nucleons and the ratio was fixed by events in the early Universe. This tells us that at some early time the Universe had to be very hot -- this was the radiation-dominated era mentioned earlier in the lecture.
In the first 10-32 of a second, conditions were so extreme that our physics is very challenged to understand them (this should not be a surprise!!).
All of physics is consistent with there only being four types of force. The weak and strong forces have to do with holding particles together in the nuclei of atoms and other things on that scale. We have already met the gravitational and electrical forces. Physicists dream of a theory that would show how all four forces collapse into just a single force law at the very high densities and temperatures in the early Universe.
Under conditions of extreme temperature and density, the forces look more and more like different manifestations of a single force law. Physicists have successfully developed a theory that unifies the strong, weak, and electromagnetic forces, called (with a little puffery) the Grand Unified Theory or GUT. The three forces would have been unified under the conditions in the first 10-35 seconds. Physicists assume that gravity can be unified under the extreme conditions in the first 10-40 seconds, but so far this has not been demonstrated.
Thus, in the first tiny fraction of second, physics was actually simpler than it is now because there were at most two force laws to deal with -- gravity and the unified weak, strong, and electromagnetic forces. This "simple" physics gives scientists confidence they can understand things even though conditions were incredibly beyond our experience!
Creation of Nucleons from Radiation
In the radiation-dominated era, photons could spontaneously create matter -- matter/anti-matter pairs would be formed. Let's look at the temperatures required to form protons:
A convenient way to express the energy of an average photon is
kT where k is Boltzmann's constant (1.38 x 10-23Joules/°K)
The temperature is
By the same type of calculation, electrons require T~6x109°K for their formation.
The approximate time when protons were formed is ~10-6 seconds at the T computed above. Neutrons can also be formed. Note that both nucleons and their anti-particles were formed and so the Universe was a hot soup of photons and nucleons as well as electron and positrons that were forming and disappearing. As the Universe continued to expand, a slight asymmetry in the numbers of particles and anti-particles arose. The temperature dropped to the point that the spontaneous creation of nucleons stopped. The number of protons continued to rise and the number of neutrons dropped because neutrons can spontaneously decay into protons and electrons. The Universe continues to expand with formation of electrons stopping when the temperature drops below 6x109°K. This happens at time~1 sec.
Primeval Nucleosynthesis
Up to time~300 seconds, the Universe is hot enough for the barrier potentials of protons to be overcome. Because the early Universe contains a significant number of neutrons (their half-lives are ~13 minutes), He can be formed by a set of reactions that cannot occur in the Sun's interior because it doesn't have any free neutrons:
Just as these reactions were ceasing, other reactions where Be was formed from 4He and Li was formed by decay of Be occurred. Only small amounts of these elements could form before the expansion of the Universe dropped the temperature below the level where nucleosynthesis could take place. Nucleosynthesis ceased at about time=300 seconds.
The relative amounts of deuterium, various isotopes of He, Be, and Li formed in the Big Bang depend on the temperature at a given pressure and hence size of the Universe. It is possible to compute the abundances in terms of the density of matter in the Universe now. By observing the abundances of these isotopes, we can determine the amount of matter that must be present now. Such measurements imply a density of 2-5x10-28 kg/m3 or qo~0.1, much less than the critical density to close the Universe.
Note that the isotopic measurements are fraught with various problems such as the fact that nucleosynthesis in stars can alter the relative amounts of these isotopes, especially 2H (deuterium) and Li.
Also note that the ratio of the abundance of 4He to H is relatively insensitive to conditions in the Big Bang. The observation that this ratio in the oldest stars is virtually identical to that expected from the Big Bang is another strong piece of evidence in favor of the Big Bang.
The Universe continued to expand and cool -- eventually the atomic nuclei captured electrons and the Universe became transparent with the emission of what is now the 3°K CBR.
The clock continued to tick, the Universe continued to cool, and at some point (a billion years after the Big Bang???), galaxies began to form.
Problems with the Big Bang Model
The Horizon Problem: The CBR that we observe from opposite sides of the sky comes from regions that were 1000x closer together when it was emitted. These regions were nonetheless millions of light years apart when the CBR was emitted. But the Universe was only 300,000-500,000 years old. How could the temperature have been so uniform if these regions were much too fart apart to have equilibrated since a light could not have traveled from one side of the Universe to the other? 
The Flatness Problem: Controversy still surrounds the exact value for the amount of matter in the Universe or equivalently, for the value of qo. However, we do know that it lies with a factor of 20 of the flat Universe value. Most models of the earliest stages of the Big Bang show that if a clump of matter formed, the Universe would rapidly have coagulated around it and we would have a closed Universe. Similar arguments imply that if the Universe developed negative curvature (e.g. an open Universe), it would have quickly developed a very strong negative curvature. Either the Universe should be exactly flat or extremely far away from flat.
Both of the previous two issues can be mitigated by invoking inflation -- a period when the Universe expanded very rapidly in size. This occured at the 10-35 sec point and the Universe expanded in size by about a factor of 1050 to become about the size of a grapefruit. This happened because space could actually repel itself while several of the fundamental forces were still acting as one.
The Structure Problem: The Big Bang makes no predictions that lead to the filamentary supercluster/void structures that we see in the Universe now. How did they develop? The leading model now is that dark matter governs the formation of structure.
Wherever small increases in the density of the dark matter occurred, the gravitational attraction made them grow into mass concentrations, carrying the normal matter with them. Prior to recombination, these concentrations were heated by absorbing photons and they re-expanded (leaving only the weak structure seen by COBE, Boomerang, and Maxima-1). However, after recombination the concentrations could cool and attract more matter. The stars, galaxies, and planets formed in these gravitational wells.
Exactly how this happened, we do not know -- but we know it did because we see galaxies in the HDF at about 1 billion years into the life of the Universe. The gap in time when matter in the Universe was converted from a nearly uniform distribution to its clumpy state at z = 5 is sometimes called the "dark ages" because we know so little (and because the cosmic background cooled to where we could not see it and stars had not yet formed in enough numbers to light it up). The simulation below (from N. Gnedin) shows one version of how the Universe emerged from the dark ages. Time starts at z = 20 and runs to z = 4.
The lower left is the gas density - dark is low density and bright high. The upper left shows how much of the gas is neutral (pink) and how much ionized (green for partial, black for total ionization). The lower right is the gas temperature, the brighter the hotter. The upper right tracks the rate of star formation.
As the time goes on, you will see how first stars (which are not visible in this movie) blow up bubbles of ionized gas (yellow to black colors), how these bubbles merge, and how the universe becomes ionized (black) and transparent.
We can look at the same process in more detail in terms of the actual distribution of gas and formation of matter in the simulation below:
It shows the growth of structures due to the dark matter gravity, then the first galaxies form and begin to merge as they collide with each other.
Dark matter places a critical role in the process of galaxy formation. Galaxies appear to have huge distributions of dark matter around the visible stars, as we saw from their rotation curves.
These dark matter halos increase the effective size of the galaxy and make it far more likely to collide with other galaxies. Starting with small scraps of galaxies, it is possible that large galaxies grow by collisions and mergers.
It is currently hotly debated how big a galaxy could form on its own and whether the big galaxies were formed primarily by assembling small ones or formed separately and just grew modestly by "eating" small ones, so the details of this picture may change as we learn more.