Chapter 3. The Thermal History of the Universe

Pair Production and Particle Annihilation

In this chapter we will discuss the rapid evolution of the Universe in its very early stages. Most of this discussion will take place when the Universe is less than 2 seconds old. The basic framework for describing the thermal evolution of the Universe uses the principle of mass-energy equivalence, E = mc² ; mass and energy are the same thing and can be converted from one form to the other. More specifically, the following reactions are possible in nature:

photon + photon --> particle + antiparticle

particle + anti-particle --> photon + photon

The top reaction represents the case of pair production. In this situation, two energetic photons collide to produce a particle plus its anti-particle. The bottom reaction represents the case of particle annihilation in which a particle plus its anti-particle meet and annihilate one another to produce energy (photons). In theory, this is the concept behind matter-anti matter propulsion. The problem is that there is little anti-matter left in the Universe, a subject to which we will later return.

In the early Universe, pair production and particle annihilation were occurring all the time. The key point, however, is the following: since the temperature of the Universe is steadily falling (due to its expansion) and since the average photon energy depends on the temperature, then the average energy per photon declines as the Universe expands and cools. The kinds of particle/anti-particle pairs that can be created depends strictly on the photon energy. For instance, the neutron has a rest mass energy given by m_nc², where m_n represents the mass of the neutron. A neutron plus anti-neutron can only be created if the initial energy of the photon is in excess of this rest-mass energy. This is why very high temperatures are required for pair-production and why, therefore, it can only occur in the very early Universe.

Figure 3.1 Distribution of photon energies. The average photon energy (peak of the curve) depends only on temperature. Photons with wavelengths less than the wavelength at which this peak occurs have energies above average.

At any given time in the Universe, the distribution of photon energies is like that of a blackbody curve (see Figure 3.1). The peak in that curve depends upon the temperature but there are substantial numbers of photons with energies greater than this average value. Nonetheless, we still we have the general rule that if

E_photon < m_pc²

then that photon has insufficient energy to spontaneously convert itself to a particle plus anti-particle of mass m_p .

What is a particle?

Most of us are familiar with electrons, protons and neutrons as being what makes up atoms. But the world of particle physics is a bit more complex than this. However, the big unknown in all of particle physics is the process that gives the particles the masses they have. Thus, while it's possible to ask - "What is a proton?" and get a sensible answer, it is not possible to ask and answer "how come the mass of a proton is 1.67 x 10^-24 grams?" In discussing the early Universe we will refer to the following kinds of particles:

1. Hadrons: A hadron is any particle which is subject to the strong force. Generally these are "heavy" particles (they have high rest mass energy). All hadrons are composed of quarks. Those hadrons that are composed of 3 quarks are called baryons. Protons and neutrons are familiar forms of baryons. Hadrons that are composed of two quarks are called mesons, which are a relatively exotic form of matter.

2. Leptons: These are light particles with low rest mass energy. They are not subject to the strong force and they are not composed of any sub-particles (there are no leptonic-quarks). A familiar example of a lepton is an electron. Another example is the neutrino, discussed below. The neutrino is a very important particle in the early Universe.

3. Quarks: These are the fundamental building blocks of hadronic matter. There are 6 known quarks: up,down,top.bottom,charm, and strange. A baryonic particle such as a neutron or a proton is made of 3 quarks. Some quarks carry charges, particularly the up and down quarks. An up quark has a charge of +2/3 while a down quark has a charge of -1/3. A baryonic configuration composed of two up quarks and one down quark would have a charge of +1. This configuration we call a proton. If one of the up quarks changes to a down quark, the charge will be 0. Now we have a neutron. In symbolic form this would be expressed as

Proton = UUD
Neutron = UDD

The remarkable thing about quarks is that if you try to take one out of a proton, for instance, to produce a particle that therefore has only 2 quarks, you can't. The force that holds the quarks together increases with increasing distance of the quarks. As you try to pull the quarks apart, their attractive bond increases. This is essentially the way the strong force works. It's called the strong force because it's very strong and allows atomic nuclei to stay together. That is, a Carbon nucleus has 6 protons in it. Why doesn't the nucleus fragment because the 6 protons with like charges all repel one another? The answer is the strong force. If you can confine protons to small spacings the strong force takes over and binds them together. Separating them requires a great deal more energy than can be supplied by simple electrostatic repulsion.

Figure 3.2 Schematic representation of a proton with 2U quarks (U) and 1D quark (D). The force between the quarks is provided by the gluons, represented as solid black balls on the force line arrows.

4. Gluons: These are mysterious sub-particles that are the carriers of the force between the quarks. There name is apropro as they literally are the "glue" that holds quarks together. As you try to separate the quarks, thus separating the gluons, the force provided by the gluons strongly increases. This is schematically shown in Figure 3.2.

5. Neutrinos: These are very light particles but there are lots of them in the Universe. They carry no charge and have a very small cross section for interaction with other forms of matter.

6. Antimatter: Antimatter has all the properties of normal matter but they are just reversed. For example, an anti-electron has exactly the same mass as an electron but has positive charge (and hence is called a positron).

Conservation Rules

Although the early Universe may appear to be quite chaotic, with matter and energy constantly changing forms, there is a set of conservation laws that strictly govern the kind of particle interactions which can occur. Specifically, there are three things that need to be conserved in any particle interaction. These are

Baryon Number
Lepton Number
Charge Number

Whatever these three numbers were in any initial state, they must be the same in a final state. But what do these numbers represent?

The Baryon number represents the initial baryon state. The units are +1, 0 , -1. A normal baryon, like a proton, would have a baryon number of +1. An anti-baryon would have a baryon number of -1. If the particle wasn't a baryon, its baryon number would be zero. Similar arguments hold for charge and lepton number. It is the strict adherence to these conservation rules that, for instance, allow for the spontaneous creation of some forms of antimatter during some particle decays, as in the case of neutrons which will decay with a 1/2 life of 900 seconds if they are not bound in an atomic nucleus.

Figure 3.3 Visualization of the decay of the neutron to its 3 decay products. The baryon number, lepton number and charge for each particle is indicated.

The decay path of the neutron is shown in Figure 3.3. Let's examine each of the decay steps in detail. The initial state is just that of a neutron. A neutron has a baryon number of +1, a lepton number of 0 (because its not a lepton), and a charge number of 0 (its neutral). Whatever the form of the decay products, their total baryon, lepton and charge numbers must sum to the initial condition:

Step 1: The neutron decays into a proton. The proton has baryon number = +1, lepton number = 0, and charge = +1. Now we have an immediate imbalance in the charge. Since conservation of charge is required (in order for the Universe to remain electrically neutral) we must balance this positive charge with a negative one. We cannot satisfy this condition with any negatively charged baryon because the total baryon number would be 2 instead of 1. Hence we require a negatively charged lepton.

Step 2: An electron is a negatively charged lepton. It has baryon number = 0 (because it's not a baryon), lepton number = +1, and charge = -1. Now we have balanced the charge but to do so required the creation of a lepton. So now we have a lepton number of +1 on the decay side when the initial value was zero.

Step 3: To satisfy the conservation laws, we need to create one more particle. This particle must have baryon number = 0 and charge =0. But it also must have lepton number = -1 and hence we must create anti-matter. A particle with these properties is the anti-neutrino. It is electrically neutral, it's not a baryon, but it is anti-matter and so its lepton number is -1.

This is just one specific example but serves to illustrate the general rules. With our understanding of the conservation laws we can now see why a photon is a special particle. Consider when, say, a proton and an anti-proton annihilate. Since the anti -proton is a mirror image of the proton, then the combined baryon, lepton and charge numbers of the proton anti-proton pair must each be zero. Hence, the decay products must be 0. A photon is a unique particle in that it's neither a baryon or a lepton and it has no charge, so its value is zero. In that sense, a photon also acts as its own anti-particle. Because a photon is neither a baryon or a lepton it has no rest mass and hence can travel at the speed of light. Only photons or "photon-like" particles have this property.

The Universe in its first 10^-43 to 10^-11 seconds: Quantum Fluctuations

The physics of the very early Universe is quite unknown. At these early times the Universe was in a very extreme state with a temperature of approximately 10³² K and a density of 10⁹⁷ grams per cubic centimeter. These are absurd numbers of course and we have no real clue what kind of physics is occurring here nor even what the meaning of "space" is under such extreme conditions. Among the unknowns is whether or not quantum mechanics exists at these early times. If it did, there may have been important consequences whose philosophical nature are like Democritus opined 2500 years ago. One of the fundamental tenets of Quantum Theory is known as the uncertainty principle. One of the variations of this principle tells us that we can never precisely know the energy state of a particular particle and the duration of that energy state. In equation form, this is expressed as

where h is known as Planck's constant and is one of the fundamental constants of nature. This is the same constant that determines the energy of a photon:

While Planck's constant is such a small value that the energy fluctuation equation above has no relevance in the macroscopic world, in the extremely early Universe, the time interval Dt could be arbitrarily small and hence the energy fluctuations DE could have been arbitrarily large. In this sense, the Universe has no choice but to happen as the energy fluctuations which produced it initially were naturally occurring.

We don't know if this is the correct physical explanation or not. On the other hand, these quantum fluctuations in the early Universe certainly allow for the spontaneous creation of very massive hadronic pairs (because the energy levels are so high). Now its most likely that these heavy hadron + heavy anti-hadron particles just annihilated one another. However, there is a possibility that some small residue of very massive hadrons are left over. If so, they could be a major mass constituent in the Universe. These particles would be weakly interacting, meaning that they would not influence the dynamics of the early Universe or decay into smaller mass particles. Weakly interacting massive particles have been given the nickname WIMPs. If they really exist, there may be several of them whizzing through your body as you read this.

The Quark-Gluon Plasma

While physicists are uncertain if WIMPS actually exist or not, there is a similarly strange state in the early Universe that most agree must have existed. When the Universe was very small, the quarks were very close together. Recall that the force between quarks , provided by the gluons, increases with increasing quark separation. When the quarks are very close to one another, the force drops dramatically. In fact, in this early Universe the quarks were so close to one another that no hadrons could exist. Hence, even though U and D quarks were present, they could not form a proton because the quarks cannot be bound together by the gluons and so the quarks had to be "free" in the sense that they were not contained in any hadron. At this point the Universe consists of a sea of high energy photons with individual quarks and gluons floating about.

The basic condition for the free quark era is that the density of the Universe has to be in excess of nuclear densities (10¹⁴ grams per cubic centimeter). This condition is fulfilled when the Universe is less than 1 millionth of a second old. After this time, the expansion has lowered the density to below the nuclear density threshold and the quarks can now be assembled into hadrons. After this assembly, conventional physics more or less holds from here on out. Prior to this, the physics of this kind of free quark/gluon plasma is largely unknown.

Unification of the forces

There were other strange attributes of the Universe at this time. Easily the most theoretically spectacular possibility in this very early Universe involves the unification of the forces. For instance, at this time the gravitational force and the strong force have the same amplitude. Why is this so? As you know, the gravitational force law goes as R^-2 but at very early times, R is extremely small (smaller than any particle) so the gravitational force is extremely large. This is a very strange physical regime which leads to the following theoretical possibilities:

particle creation could come directly from the gravitational field.
all of the Universe could be in the form of one particle but that particle would be indistinguishable from the gravitational field of the Universe.

As the Universe is rapidly expanding, R is rapidly increasing and the gravitational energy is going down so that the gravitational force becomes much weaker than the strong force and the two become separate forces. By the end of this era, (10^-11 seconds), a similar separation occurs between the electrostatic force and the weak force. Whenever these symmetric forces are decoupled the Universe is said to go through a phase transition. The symmetry breaking is done in a specific manner so that the forces have now separated and are defined by a characteristic ratio of their strengths. This defines the physics of our Universe. Much theoretical research is currently underway to better understand what causes the symmetry breaking and allows the forces to assume their characteristic values. Thus another question we are not allowed to ask is "how come gravity is as weak of force as it is?". We just don't know.

The Matter/Anti-matter Asymmetry

Between 10^-11 and 10^-2 seconds the Universe continues to cool rapidly. The reaction rate (the conversion of energy into mass and the annihilation of matter anti-matter pairs) was so large that the Universe was governed by a statistical process known as thermodynamic equilibrium. Under conditions of thermodynamic equilibrium the only thing that matters is the temperature. Hence, the conditions at any one time in the early Universe do not depend on the conditions that happened just before that time. The only thing that determines the physics is the temperature at a particular epoch. This is a large simplification which allows us to identify statistical processes that might have been at work.

In so doing, however, we run into another important unknown. Think about it- if the Universe had exactly equal amounts of matter vs anti-matter then it would have totally annihilated itself and the Universe would only consist of photons (energy) and no matter. Yet you are made of matter so what gives? One possible explanation is that some agent in the early Universe acted to separate out the matter and anti-matter into different areas thus preventing interaction and annihilation. However, if that was the case, we would expect to observe entire galaxies made of anti-matter in the same way that we observe galaxies made of matter. No such anti-matter galaxies have ever been detected.

Hence, we are driven to another conclusion which is that there must have been an excess of matter over anti-matter. But how big is this excess? Well, we can measure that directly by comparing the energy density in the microwave background with the matter density in the Universe. This gives us our canonical photon-to-baryon ratio of about 1 billion. So for every one billion anti-quarks that existed in the early Universe, there was 1 billion and one quarks. This small asymmetry is the matter residue that survived the early Universe to produce all the stars, galaxies and observers today. However, this leads to another fundamental question that has no answer yet: what produced this asymmetry and why is it roughly one billion to one, instead of, say, one trillion to one, one million to one, etc, etc? Again, we don't know the physics that produced this exact asymmetry although there have been some suggestions that the particular form of the electro-weak symmetry breaking produced this asymmetry at a preferred value.

Epoch 1: T = 0.01 seconds

At this time the temperature has cooled to about 100 billion degrees and the density has dropped to around 1 billion grams per cubic centimeter. These conditions are still more extreme than that found in any star. The constituents of the Universe at this time are:

Protons, neutrons, electrons and neutrinos
One billion photons for each proton, neutron, and electron
matter and energy exist in thermal equilibrium

At this time the Universe is still so dense that it is opaque to neutrinos. Ordinarily, neutrinos do not easily interact with other forms of matter, however, in cases of high density, the neutrinos can not travel through the matter without interacting. There are two fundamental neutrino interactions which are occurring

anti-neutrino+proton <----> neutron+anti-electron

neutrino + neutron <------> proton + electron

In these two reactions we can see that neutrinos are the mediating agent which allows protons to be converted to neutrons and vice-versa. A good exercise for the student at this point is to verify that the above reactions indeed follow the conservation rules established previously.

The above reactions keeps the number density of protons and the number density of neutrons constant. That is, there is thermal equilibrium between protons and neutrons. This is extremely important and will lead to a testable prediction later. Recall that a free neutron will decay in about 900 seconds (its half-life). At t = 0.01 seconds, none of the neutrons have decayed and furthermore, new neutrons are being made from protons. So, as long as the neutrinos can interact with neutrons and protons, this balance will be kept. Hence the proton-to-neutron ( p/n ) ratio is 1.

Epoch 2: time = 0.11 seconds

The Universe has now cooled to 30 billion degrees but its density has dropped even more, down to around 10 million grams per cubic centimeter (the density of a white dwarf star). Now we see the first change in the p/n. Since neutrons are slightly heavier than protons (by the mass of an electron) then the n --> p cycle in the previous neutrino reaction is favored over the p --> n cycle. So we have a small loss of neutrons which increases p/n to 1.6. Still, the Universe remains in thermal equilibrium between matter and energy but the reaction rate is rapidly slowing down due to expansion and cooling. Soon, no more matter will be created from pair production and then the total amount of matter and energy is fixed forever, in the expanding Universe.

Epoch 3: time =1.1 seconds

The temperature now is 10 billion degrees (still hotter than centers of stars) and the density is around 1000 grams per cubic centimeter. At this point we have a very significant event. The neutrinos now decouple from their interaction with matter because the density has lowered to the point where the matter becomes transparent to the presence of neutrinos. Since the neutrinos no longer are interacting with any other matter in the Universe they can free stream to fill the Universe. This is another difficult concept for students to understand so consider the following analogy.

Suppose that you are in your astronomy lecture room on campus. The outside of that room represents the Universe. Now suppose you are the only student in that room but you are surrounded by a network of kangaroos with boxing gloves jumping up and down and flailing their arms. As you try to make your way out of the room, you encounter one of these renegade roos who randomly alters your direction of motion, until you encounter another roo who again randomly alters your direction. Assuming you can survive the renegade roo punches, you should realize that you are not getting out of this room anytime soon. Thus, you are opaque with respect to the network of roos. Only when the roo density drops can you exit the room and "stream" out into the Universe. As we will later see, the Universe is also opaque to its own radiation for quite sometime as the radiation is coupled to the matter just like you were coupled to the roos in the above analogy.

After the neutrinos decouple from the matter and fill the Universe, they form a neutrino background (an analogous decoupling later on will form the photon background which we now measure as the CMB). In principle, there is a cosmological background of neutrinos which is impossible to detect with today's technology. If neutrino detectors improve in the future and this background is detected, it will provide further (and very strong) support for this model. The ratio of neutrinos to CMB photons is about 1/4, so on average, there are 100 cosmological neutrinos per cubic centimeter everywhere in the Universe, including where you are right now.

The second major event that occurs during this epoch is the end of particle creation. When the Universe is 10 seconds old, it has cooled to the point where the average energy per photon is less than the rest mass energy of any known particle, hence no particles can be created. However, in this interval of 1-10 seconds, the Universe is already below the threshold energy for the creation of protons and neutrons. Only very light particles like electrons (which have a rest mass energy 2000 times less than that of a proton or neutron) can be created in this window. This leads to a very interesting situation as the reaction electron+anti-electron --> photon+photon is now greatly favored over the reverse reaction photon+photon --> electron+anti-electron. In fact, it is this electron + anti-electron annihilation that produces most of the photons that we now observe in the CMB. But, this extra "re-heating" of the Universe is not experienced by the neutrinos who have now fled the scene. Thus, the neutrino background is also colder (of lower energy) than the 2.74 K background we now measure for the CMB.

Since the neutrinos are gone then the p/n ratio continues to increase. By t = 10 seconds this ratio is now 3:1. No more particle creation is occurring from the photon field. The Universe consists of protons, neutrons and electrons plus a photon background and a neutrino background.

Epoch 4: T = 14 seconds to 3 minutes

At the start of this epoch, the temperature of the Universe has cooled to 3 billion degrees and the density has dropped to 100 grams per cubic centimeter. No more particle creation is occurring and the abundance of electrons and anti-electrons now is constant. Remember, these are the lightest particles that we know of and hence have the longest formation window. All these electron and anti-electron pairs now annihilate thus producing the final photon background with the characteristic that there are one billion photons for each particle.

During this epoch, however, stable atomic nuclei (e.g. Helium) can't form yet because the Universe is expanding very fast and is still filled with very high energy photons. Thus, the neutrons continue to decay and p/n continues to rise. At the end of this epoch p/n is 7 and the neutrons are starting to decay away completely. Clearly, if all the neutrons created in the early Universe were allowed to decay, then we would have no periodic table of elements and hence no life in the Universe. For life to therefore arise, some agent must intercede to prevent the decay of the free neutrons. Fortunately, a natural mechanism arises.

Epoch 5: 3--15 Minutes and The Formation of Light elements

The Universe has now cooled to a temperature of a few hundred million degrees. The density is approximately 10 grams per cubic centimeter. The conditions of the Universe (temperature and density) are now very similar to the conditions inside a star. However, since the Universe is still expanding rapidly, the Universe is not like a star in the sense of being a stable place where the temperature and density remain constant. At this time p/n is 7 and the first steps of the proton-proton cycle begin. This is exactly the same thermonuclear fusion cycle that the Sun uses to generate its energy.

The first step in the cycle is the fusing of two protons to make deuterium. Deuterium is an element with two nucleons, a proton and a neutron. It is hydrogen with an extra neutron in its nucleus and therefore is an isotope of hydrogen. We will refer to deuterium with the symbol ²H indicating that its hydrogen with 2 nucleons ( ¹H is just a proton). In symbolic form, the reaction is

H + H ==> ²H + positron + neutrino

However, deuterium is a very fragile nucleus and can easily be broken apart by a high energy photon:

²H + photon ==> p + n

and the neutron created in this way will decay unless it can bind with another proton to form an atomic nucleus.

This competition between the creation of ²H via fusion and its destruction via photo-dissociation sets up an interesting race condition. Will deuterium combine with another proton to make a nucleus with 3 nucleons or will it be photo-dissociated before it can do this? This race condition depends on the density of protons:

if the density is high then deuterium will fuse with another proton to make ³He
if the density is low then most of the deuterium will be photo-dissociated before making ³He
a high density Universe means a low density of deuterium
a low density Universe means a low density of helium because most of the deuterium is destroyed before it can be fused into helium.

Because of this, if we can measure the deuterium and helium abundances in the Universe, we can get a handle on what the proton density was initially. Measuring the helium abundance is significantly easier than measuring the deuterium abundance and so we have the most observational data for that.

The next step in the proton-proton chain is

²H + H ==> ³He + photon

³He is a new element that has two protons in its nucleus. Each element in the periodic table has a unique number of protons in its nucleus. Isotopes of those elements are formed whenever there are different numbers of neutrons (could be more, could be less) in the nucleus. The most common form of helium is ⁴He: 2 protons + 2 neutrons. Helium with only one neutron in its nucleus is still helium (because there are 2 protons). So ³He is an isotope of helium. Once ³He is formed, the next step of the reaction can occur fairly quickly because ³He has a relatively high binding energy and is not susceptible to photo-dissociation.

The final step in the proton-proton chain is

³He + ³He ==> ⁴He + p + p

where we now have formed stable ⁴He. If it were not for the formation of ⁴He, then the Universe would be devoid of neutrons. Thus the free neutrons created in the early Universe end up in ⁴He nuclei.

The Predicted Helium Abundance

Prior to the epoch of nucleosynthesis, the p/n ratio was 7. So for every 14 protons there are 2 neutrons. The end result of the proton-proton changes is the conversion of 2 protons and 2 neutrons into 1 ⁴He nucleus. Hence, after the reaction our initial mix of 14 protons and 2 neutrons has been changed to 12 protons and 1 ⁴He nucleus. The mass of ⁴He is approximately 4 times the mass of a proton. This leads to a very specific prediction for the mass fraction of the Universe which is in the form of helium:

where the masses in the above equation are all in units of proton mass. The prediction that the helium mass fraction is 25% has been confirmed many times via observation. If this were not the case, we would have a severe blow to our model. The good agreement between the prediction and the actual observations provides further support of the Hot Big Bang model.

Overall, the abundance of helium is sensitive to three parameters:

The p/n ratio at the time that nucleosynthesis starts
The ratio of photons to baryons
The actual baryon density

The good agreement between theory and observation suggests that our estimate of these quantities can't be too far wrong.

Elements beyond Helium

In addition to making ⁴He , there is some limited nucleosynthesis that acts to build heavier elements. These reactions are

³He + ⁴He ==> ⁷Be + photon
⁷Be + electron ==> ⁷Li + photon
⁷Li + proton ==> ⁷Li + neutrino
⁷Be + proton ==> ⁸Bo + photon
⁸Bo ==> ⁸Be + anti-electron + neutrino
⁸Be ==> ⁴He + ⁴He

In the above reaction sequences one can see that the usual end product is a return to ⁴He. However, these reactions don't necessarily go to completion (because the Universe is expanding) and so trace amounts of ⁷Li (Lithium) should also exist. Observations have tried to detect these trace amounts in the oldest stars in our Galaxy with some limited success. The observed abundances agree well with the predictions of Big Bang nucleosynthesis.

However, this element production stops at ⁸Bo (Boron) which has 5 protons and 3 neutrons in its nucleus. Heavier elements cannot be produced because:

⁸Bo is very unstable (1/2 life = 0.8 seconds) and rapidly decays into the even more unstable element ⁸Be (1/2 life ~ 10^-17 seconds) which rapidly decays back into two ⁴He nuclei. This short half life makes it impossible for either Boron or Beryllium to capture a ⁴He to make Carbon.
The Universe is cooling too fast for the triple-alpha reaction to occur. Namely
⁴He + ⁴He + ⁴He ==> ¹²C

does not occur. We are now done with element production. At this point the Universe consists of photons and the elements H,²H,³He,⁴He,⁷Li as well as one billion photons per hydrogen atom. At this time these photons all have energies sufficient to ionize hydrogen and helium. Thus, in addition to a sea of photons, a sea of free electrons also exists and this has a very important physical effect.

The Epoch of Decoupling

When the Universe is 15 minutes old, the temperature has now decreased to 1 million degrees and thermonuclear fusion stops. However, the energy per photon is still very high so that no neutral atoms exist. The Universe is completely ionized at this point. It is an expanding sea of mostly photons in which are mixed small amounts of atomic nuclei and free electrons. There is a physical mechanism, called electron scattering, which now couples the distributions of matter and radiation, rendering the Universe opaque to its own radiation. This method works as follows (see Figure 3.4).

Figure 3.4 Visualization of photons (gray wavy lines) having their direction of motion changed when they encounter an electron. This is electron scattering.

A photon that encounters a free electron will have an interaction with the electron because it has an electric field. This interaction takes the form of scattering. Scattering randomly alters the direction of motion of the photon in the same way that the renegade Kangaroos randomly altered the motion of the student discussed earlier. There is no photon energy loss in this process, the photon merely changes its direction of motion until it encounters another free electron and is scattered again. So each individual photon will experience multiple scatterings by this network of free electrons. There is essentially no way the photons can escape from this network; they are constrained to bounce around inside of it. The only way to escape is when the free electrons disappear and become bound to atomic nuclei. This will only occur when the Universe cools to the point that it is no longer ionized. For the moment, we have 1 billion ionizing photons per electron and so every time this reaction tries to occur:

proton + electron ==> H-atom

it is immediately undone by this reaction

H-atom + photon ==> proton + electron

The matter and radiation are in thermal equilibrium so the matter temperature is also a million degrees and therefore it will not easily clump. Moreover, there is momentum transfer or radiation drag produced on the matter by the radiation field so every time two matter particles try to get together, the process is interrupted by the radiation field. This is schematically shown in Figure 3.5. This leads to a big problem. How is the Universe ever going to grow a galaxy if matter can not clump together to start the process?

Figure 3.5 Visualization of two mass particles being bombarded by radiation and thus unable to gravitational "stick" together. The red arrows indicate the random directions of momentum the particles get when they are struck by the radiation. The arrows represent gravitational attraction.

The tight coupling between the radiation and matter means that the matter should be distributed like the radiation and we know, from observations of the CMB, that the distribution of radiation is very smooth. If the matter distribution were to become completely smooth, then there would be no collapse into individual clumps which could grow into galaxies eventually. We will examine this dilemma more in Chapters 4 and 5 because it is somewhat challenging to understand. For now, we conclude this Chapter by explaining how radiation and matter become decoupled.

The coupling between matter and radiation will continue as long as the Universe remains ionized. Recall that in Chapter 2 we showed that the energy density in the radiation field naturally decreases much faster than the energy density in the matter field. Eventually this means the Universe will cool to the point where the electrons can recombine with the protons. When this happens, the scattering network of free electrons is now gone and the radiation has no mechanism to couple to the matter.

Thus, there is some surface of last scattering in the evolution of the Universe from which the photons are finally no longer scattered but free stream to fill the Universe. This creates the CMB signal that we observe. The observed signal is greatly redshifted from the time of its creation when the temperature was approximately 3000 K (see below). Prior to this time, no external observer (e.g. us) could get any signal because the photons were still being scattered around. Perhaps the following thought experiment can help to better understand the relation between the surface of last scattering and information received by detectors.

Suppose you go to a baseball game and sit in the center field bleachers. Under most circumstances you have a clear view of the pitcher and the batter. The pitcher throws a pitch to the batter who hits a home run right at you. What do you observe? You observe the source of the baseball that you are about to catch as being the bat swung by the hitter. Nothing has interfered with the production of the home run and its trajectory to you. Now, imagine that the baseball field is filled with a network of indestructible windmills. When a hit baseball strikes a windmill, the direction of the baseball's motion will be randomly altered (we assume the baseball is also indestructible). The scattering network is so thick that you can't see the pitcher or the batter. In fact, you will never even see the baseball until it emerges from the last row of the windmills because only then can the baseball be scattered towards the bleachers. Hence, the last row of the windmills is the last scattering surface and the observer only sees baseballs (photons) emerging from this source.

Figure 3.6 Visualization of scattering. See text for a description.

This concept of scattering is shown as an artist conception in Figure 3.6. Here we see a detector (the eyeballs on the horizon) with some photons (blue and red balls) heading towards it. Imagine the source of the photons has a cannon shooting balls. Between the cannon and the detector is a scattering network (windmills in this case). As the photons enter into the scattering network their direction of motion changes and few of them will actually reach the detector. If the scattering network is sufficiently thick, then you can not see the source of the photons (the cannon) but would only observe photons coming out in random places out of the network. If you image the space outside the network as the Universe, then its easy to see that scattering will produce a homogeneous distribution of photons. If we shoot enough cannon balls into the scattering network, a ball will be scattered to every point in the "Universe".

Of interest to understanding the overall evolution of the Universe is the actual time it takes for the electrons to recombine with the protons so that radiation and matter become decoupled. This depends on two things:

1. The ionization energy of hydrogen ( 13.6 electron volts (eV) )
2. The ratio of photons to protons

Figure 3.7 Distribution of photon energies for a temperature of 3000 K. The point marked A represents the average energy per photon. The point marked 13.6 represents the ionization energy of hydrogen. All photons with energies greater than this will ionize hydrogen. Photon energy is increasing to the left in this curve. Even though only a small percentage of the total number of photons is contained in the region of the curve above 13.6 eV, the large ratio of photons to protons means that the matter can still be ionized at this temperature.

For the conditions of our Universe, this time is approximately 100,000 years and we can understand this timescale as follows: Normally, a temperature of 50,000 K is required to ionize hydrogen (e.g. like an H II region in our galaxy). At this temperature, the average energy per photon corresponds to the ionization energy of hydrogen. However, in the early Universe, there are many more photons than protons so even at lower temperatures, there will be still enough ionizing photons to continue the ionization. This is why the timescale depends on the ratio of photons to protons. If that ratio were larger, the timescale would be longer and the tendency to produce a smoother mass distribution would be larger. In fact, no galaxies might have even formed if the ratio of photons to protons was significantly higher than 1 billion to one. As schematically shown in Figure 3.7, the number of ionizing photons will become less than the number of hydrogen atoms when the Universe has cooled to 3000K. This means that an electron can recombine with a proton without fear of being immediately re-ionized because the number of ionized photons is lower than the number of electron-proton pairs.

Recombination therefore occurs at a temperature of 3000 degrees. The photons now no longer interact with the matter and free stream to fill the Universe. This is what we presently observe as the 3 K microwave background. This radiation has been redshifted by a factor of 1000 by the time the Universe has aged to 10 billion years and we detect it. This also means that we cannot observe the Universe when it was younger than 100,000 years in the same way that you cannot observe a baseball from the bleachers until it has reached the surface of last scattering, even though the baseball was "created" at a much earlier time.

After recombination has occurred, radiation is no longer an influence on the distribution of matter. Hence, matter will clump around any surviving density enhancements. The question now becomes, how did those density enhancements survive the radiation dominated era, and what is their nature? Ultimately these density enhancements have to grow to produce the structure we observe today. Since photons can be temporarily trapped in these structures and either gain or lose energy as they traverse them, their signature is imprinted on the CMB. This density fluctuation signature is what COBE detected in the form of temperature fluctuations (see Figure 2.8).

We are now done with the radiation dominated history of the Universe and the next two chapters will move on the matter dominated era. There we will focus on 1) observations that help us determine the overall size scale and mass density of the Universe 2) observations that suggest the presence of large amounts of dark matter and, 3) how structure actually formed in the Universe.

Summary

Figure 3.8 Time line of the early Universe with important events noted.

This chapter has detailed the evolution in the first few seconds and minutes of the Universe. The evolution is characterized by rapid expansion and cooling. Different physical regimes can be defined depending upon the temperature and density of the Universe. Important events that occurred in this evolution are graphically represented in Figure 3.8. The first second of the Universe is characterized by the conversion of mass into energy and vice versa. The reaction rate is so large that conditions of thermal equilibrium exist. That is, the behavior of the Universe at any epoch is dependent on its temperature and not what happened prior to that time. Temperature.

At early times, the energy density of the photons was so high that you could get particle creation from the photon field and/or energy creation from the matter field.

photon+photon <=====> particle + anti-particle

The kinds of particles +anti-particles that are created depends strictly on the average energy per photon which depends only on the temperature. If the photon energy is less than m_pc² then m_p can't be created. Only particles with mass less than m_p will be created. ( m_p just represents the rest mass of some generic particle). Since a proton is 2000 times heavier than an electron, the window of opportunity for creating electrons/anti-electron pairs from the photon field is a lot longer than for creating protons/anti-proton pairs.

In the early Universe the following regimes can be defined:

In the very early Universe (prior to 10^-11 seconds), lots of strange pairs of hadronic particles could have been produced from Quantum Fluctuations. We know very little of the physics that would be operative in this regime.

From 10^-6 seconds to 0.1 second was the window of opportunity for creating protons and neutrons and other "normal" particles.

The Universe was opaque to neutrinos prior to it being a second old and this kept the abundance of protons equal to that of neutrons.

At 1 second the neutrinos escape and the neutrons start to decay. The only thing that prevents their decay to zero is the formation of atomic nuclei via thermonuclear fusion. This fusion epoch starts around t=3 minutes and lasts until t=15 minutes. The proton-to-neutron ratio at the start of this epoch is 7-to-1. This ratio makes a solid prediction that the Helium Abundance of the Universe should be approximately 25%. This value is observed.

From 15 minutes to approximately 100,000 years the Universe was radiation dominated. This means that gravity was not effective in trying to clump together ordinary baryonic matter as the radiation pressure would try to smooth out any clumps that could have formed. The existence of galaxies, therefore, may require some exotic form of matter in the Universe which was unaffected by this radiation pressure.

From 15 minutes to approximately 100,000 years the Universe remains opaque to this radiation because the radiation and matter are coupled due to electron scattering. The radiation can only free itself from the matter when the Universe cools to the point where the free electrons can recombine with the protons and the Universe is no longer ionized. This occurs at a temperature of 3000 K and is the source of the CMB radiation that we now observe.

The three big unknowns in the above sequence are:

2. What physics determined the masses of the particles? That is, why does a proton have the mass that it does?

3. What physics determined the matter-vs-anti-matter asymmetry?. That is, why for every 1 billion anti-protons were there 1 billion and 1 protons?

Once the radiation dominated Universe has ended, structure can form and the Universe that we observe today can come into existence from fluctuations in the distribution of matter. How these fluctuations survive the radiation dominated era is currently a mystery.