How is the duration of BBN calculated?

[Doctor Strange]

BBN is believed to have started a few tenths of a second after the big bang when temperatures were roughly 116 GK, and lasted until the universe had cooled to about 1.16 GK. According to the FLRW metric, this was about 16 minutes.
I get the impression that the duration is important, but don't understand how. Are the relative abundances of elements dependent on the temperature difference (i.e. BBN starts at 116 GK when stable nucleus can form and ends at 1.16 GK when they are frozen), or is it time dependent (e.g. it starts at 116 GK and goes for 16 minutes after which time no more nuclei can be formed because the raw materials are used up)? Or is it volume dependent (e.g. It begins forming at 116 GK and continues as long as the volume of space expands at some rate)?

 

[Chalnoth]

The duration is important because it changes the mix of elements: if the duration of BBN were long enough, then all of the normal matter in our universe would be made of iron atoms. The short duration makes it so that heavier elements don't have a long enough time to form.

As for BBN itself, the physics involved in BBN is well-studied in terrestrial experiments, and a combination of those physics plus thermodynamics is used to calculate the primordial elemental abundances. I believe that the more detailed estimates use fairly complicated computer models to arrive at their conclusions.

 

[Doctor Strange]

The duration is important because it changes the mix of elements: if the duration of BBN were long enough, then all of the normal matter in our universe would be made of iron atoms. The short duration makes it so that heavier elements don't have a long enough time to form.

I've seen some descriptions of BBN state that the expansion rate is a factor, but can't find a more detailed explanation. One source says:

As the temperature dropped, the neutron-proton inter-conversion rate fell faster than the Hubble expansion rate, $H ∼ \sqrt {g∗GN} T^2$, where g∗ counts the number of relativistic particle species determining the energy density in radiation (see ‘Big Bang Cosmology’ review).​

I've got a pretty good idea how FLRW metric expansion works but I'm having trouble understanding how this relates to a particle reaction. Any insight in layman's terms would be appreciated.

 

[Chronos]

When the universe was just a baby [prior to about 3 minutes old] only elementary particles [e,n,p] could exist. At first the only element was naked hydrogen nuclei [free protons], which, fortunately, is stable. Free neutrons, however, are more than happy to bind with free protons at around 3 billion degrees because they will decay into protons out of despair if left unwed for more than a few minutes. So they rapidly married protons to form deuterium and helium. As the temperature dropped to about 1 billion degrees the number of deuterium and helium nuclei apiked, and the abundance of unwed neutrons plummeted. This is called the deuterium bottleneck. By the time the universe was about 20 minutes old, the available neutrons had been depleted. The only way heavier nuclei could form was through hydrogen and helium nuclei fusing with one another in various combinations resulting in lithium, berrylium, and a small amount of boron. The universe cooled too fast for this to continue for very long and never did get high enough to synthesize the next stable element - carbon - until stars came along. The binding energies of the elements is indicative of the energy necessary to synthesize an element, which must exceed its binding energy.

 

[Doctor Strange]

The binding energies of the elements is indicative of the energy necessary to synthesize an element, which must exceed its binding energy.

This is a fine description of the sequence, but I still have this question about the timing. It appears that there's a window after the temperature has dropped below 3 billion K of about 20 minutes before neutrons decay into protons, electrons and anti-neutrinos. I've seen various sources refer to the issue of the Hubble expansion rate as one of the factors that determined how quickly the neutrons were depleted. Can you add any insight to how this works?

 

[Chalnoth]

This is a fine description of the sequence, but I still have this question about the timing. It appears that there's a window after the temperature has dropped below 3 billion K of about 20 minutes before neutrons decay into protons, electrons and anti-neutrinos. I've seen various sources refer to the issue of the Hubble expansion rate as one of the factors that determined how quickly the neutrons were depleted. Can you add any insight to how this works?

The temperature drops at a rate of 1/a, where 'a' is the scale factor. A faster expansion means the scale factor increases at a more rapid pace, which causes the temperature to drop more quickly.

If the temperature is too high, no atomic nuclei can exist, because they get destroyed as quickly as they are created. Once it gets too low, nuclear fusion stops. So there's a range in temperature where the elements are produced, and how long the universe spends at that temperature depends upon the rate of expansion.

 

[Chronos]

The Hubble expansion rate [which was huge in the infant universe]strongly affects the depletion rate of free neutrons. The universe is still dense and hot enough that arounf T=1 second weak interactions involving neutrinos can convert neutrons to protons and vice versa. Since neutrons are more massive than protons, they are less abundant — so conversion of a neutron to a proton is less probable than conversion of a proton to a neutron. At thermal equilibrium, the ratio of neutrons to protons is set by e^−Q/kT- where Q ≡ (m_n −m_p)c² = 1.2934 MeV, which occurs around T=3 seconds and kT declines to about 0.7 MeV. Since the conversion rate drops rapidly as density and temperature declines due to expansion, proton to neutron conversion has a very short window, and, the neutron to proton ratio “freezes in” at ≈ e^{[−1.2934/0.7]}, or about 1/6. The decay time for free neutrons is ∼ 900 sec, so by around T= 2 minutes when deuterium synthesis began, a non-negligible fraction of the neutrons left over from “freeze-out” have decayed. The neutron-to-proton ratio at this time is about 1/7. Had the infant universe expanded more slowly, proton to neutron conversion would have persisted longer resulting in a higher 'freeze in' ratio of neutrons to protons and synthesis of more heavy elements during BBN. Since observation constrains the mass ratio of elements heavier than hydrogen to about 25% of the total, this enables us to back fit the temperature history of the infant universe sufficient to account for neutron depletion.

 

[PeterDonis]

One source

Which source? Please give a specific reference.

 

[Doctor Strange]

Which source? Please give a specific reference.

http://pdg.lbl.gov/2011/reviews/rpp2011-rev-bbang-nucleosynthesis.pdf

 

[PeterDonis]

http://pdg.lbl.gov/2011/reviews/rpp2011-rev-bbang-nucleosynthesis.pdf

Thank you.