Creating Helium

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.

Overview of the First Three Minutes:

Observations:

The observations indicate that the universe is expanding and cooling (just like an expanding gas cools). Hence, in the past the Universe was hotter and denser than it is currently. Extrapolation to the distant past demands that the Universe was very hot and very dense.

The condition of thermal equilibrium means that the behavior of the Universe at any epoch is dependent on its 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 photon energy is less than mpc2 then mp can't be created; only particles with mass less than mp will be created.

Since a proton is 2000 times heavier than an electron, the window of opportunity for creating electrons from the photon field is a lot longer than for creating protons.

In the very early Universe, lots of strange pairs of particles could have been produced from Quantum Fluctuations --> we know very little of this physics.

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 is 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 about The Helium Abundance of the Universe which is described above.

Three big unknowns in the above are: