Question

7A. Calculations of the primordial material that would form a "quark soup" a few hours after the Big Bang show that the early universe consisted almost entirely of hydrogen (¹₁H = 76 weight %) and helium (⁴₂He = 24 weight %). The present-day atomic ratio of H/He is estimated as 12.5/1. Assuming the present universe consists ONLY of H and He, calculate the present-day wt. % of H and He.

B. Does the present-day wt. % of H and He agree with the Big Bang theory?

C. What physical mechanism(s) explain(s) why the present-day wt. % of H and He does/does not agree with the Big Bang theory?

Answer 1

Solution:

A.

Given that,

The present-day atomic ratio of H/He is estimated as 12.5/1

Weight of He is 4 times that of H.

Thus,

wt ratio of H/He is 12.5/4.

Hence, the present-day wt. % of H = (12.5/16.5 ) * 100% = 75.75 %

the present-day wt. % of He = (4/16.5 ) * 100% = 24.24 %

B.

According to Big Bang theory and using powerful telescopes, scientists have made extensive spectroscopic surveys of distant stars and galaxies. The data indicates that hydrogen and helium make up nearly all of the nuclear matter in the universe. The most abundant element, hydrogen, accounts for 74% of the mass while helium contributes 25%. Heavier elements comprise less than 1% of the total.

Hence,  the present-day wt. % of H and He does agree with the Big Bang theory.

C.

Prior to Big Bang Nucleosynthesis (B B N) Two key nucleon reactions before B B N commences are:

• a neutron interacts with a positron to create a proton plus anti neutrino (and vice versa), and
• a neutron interacts with a neutrino to create a proton plus electron (and vice versa).

Initially the temperature was high enough for all these reactions to take place, maintaining an equilibrium of protons and neutrons. However, as the temperature rapidly dropped, the neutron-proton inter-conversion rate per nucleon fell faster than the Hubble expansion rate, favouring protons ahead of neutrons. About one second after the BB, the temperature had dropped to about 0.7 MeV, too low for these reactions to continue (the "freeze-out"), by which point the N:P ratio had fallen to 1:6.

Nucleosynthesis begins

BBN was now spluttering into action, and the first multi-nucleon element formed is the simplest: deuterium (one proton plus one neutron). However, the temperature was still too high for deuterium to survive, as the energy of some photons was higher than deuterium's binding energy, and any 21H would quickly photodissociate. This period is called the "deuterium bottleneck"; but once the temperature had dropped to about o.1 MeV, the 21H could survive, and as a result there was a sudden and significant increase in the proportion of deuterium atoms present. And as helium-4 has the highest binding energy per nucleon among the lighter elements, almost all this deuterium quickly ended up as 42He.

However, free neutrons are unstable and decay with a half-life of 611 seconds, so any neutron that hadn't managed to get to "safety" within an atomic nucleus during this brief 20-minute BBN period was most likely to have decayed into a proton. As a result, the final N:P ratio ended up at 1:7.

This final 1:7 neutron-proton ratio tells us that for every 14 protons at the end of BBN, there were two neutrons. Since on average those two neutrons found themselves with two protons in a 42He nucleus, this leaves 12 protons without a partner. A proton on its own is a hydrogen ion, so we can predict that once the temperature had dropped too low for further nucleosynthesis (at around 30KeV, or about 20 minutes after the BB), there were roughly 12 hydrogen atoms for each atom of 42He.

By the end of Big Bang nucleosynthesis, on average 12 out of 13 atoms will be hydrogen (92%) and the remaining atom will be helium (8%); or by mass, with a helium atom four times heavier than a hydrogen atom, it's 4:12 or 4/16 ths or 25% helium, and 75% hydrogen. And this is, indeed, exactly what we now find.

While the question asks "How does the ratio of hydrogen to helium help prove the big bang theory?", there is even stronger support for the standard BBN model when we look at the lesser "relic" product of that brief burst of nucleosynthesis: deuterium. The production of this isotope is highly sensitive to the primordial baryon abundance, and observations of the Universe's large scale structure and of temperature fluctuations in the cosmic microwave background radiation have tightly constrained that number, enabling a well-bounded prediction for the relic abundance of D. This D/H prediction is in close agreement with the current abundance inferred from observations of low metallicity galaxies, providing even stronger support for the standard model.

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