Big Bang Nucleosynthesis

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1. BBN – neutron to proton ratio As the universe continues to cool down, the protons and neutrons start to fuse
together and form heavier elements. This era is called the era of big-bang nucleosynthesis and occurs at t ⇠ 3 minutes.
Here, we calculate the neutron to proton ratio, Xn = nn/np, at the time of BBN. Its importance will become clear in the
next problem.
For simplicity, we will assume that in the radiation dominated era, the universe expands and cools as (eq. 15.5 in Notes)
kBT(t) ⇡ 1MeV ✓ t
1 s◆1/2
. (.1)
At early times, the weak interactions that transmute neutrons to protons and vice versa (here ⌫ is an electron neutrino
and ⌫¯ its anti-particle),1
n + ⌫ $ p+ + e , (.2)
are suciently fast that protons and neutrons remain in chemical equilibrium, with the reaction rate going to the left equals
that going to the right. As neutrons are slightly heavier than protons, with a rest mass energy di↵erence of Q = 1.293
MeV, energy conservation dictates that the electron on the right needs to be more energetic, and therefore rarer, by
that amount than the neutrino on the left. At the high temperature of concern, neutrinos and electrons have the same
Maxwell-Boltzman distribution. So to balance the rates one requires the following ratio
Xn = nn
np
= exp ✓
Q
kBT
◆
, (.3)
or, the protons and neutrons are in thermal equilibrium, with their number ratio determined by their energy di↵erence
(Boltzman distribution).
The thermal equilibrium between neutrons and protons are maintained as long as the reaction rate is faster than the
expansion of the universe. If not, the ratio between neutrons and protons is frozen-in and is called the ’fossil’ value. An
earlier freeze-out means a more neutron-rich universe.
1. To compute the time of freeze-out, we need both the rate of weak interactions and the rate of universal expansion.
The latter can be obtained from eq. (.1). For the weak interaction rate, the cross-section for a neutron to interact
with a neutrino is (c.f. Mukhanov 2008, but there is a mistake in his eq. 18)
n⌫ ⇡ 2.13 ⇥ 1042 cm2
✓kBT
Q + 0.25◆2
. (.4)
This tiny cross-section, typical of weak interactions, explains why it is so hard to detect neutrinos on Earth.2 Use
this cross-section, and the number density of neutrinos as a function of temperature, to obtain the average reaction
rate (the inverse of mean-free-time) for a free neutron. Express your result as a function of y = kBT /Q. Multiply
your result by a factor of 2 to account for the second reaction.
2. Balance your newly obtained reaction rate against the rate of expansion to obtain the temperature and time at
freeze-out. What is the freeze-out neutron-to-proton ratio? Compare your answer to the more detailed calculation
which yields Xn = nn/np = 0.158, or one neutron for every 6 protons

Sample Solution

inly Chinese was a very good idea since it brought huge result to CTs.

The timeframe also had significant effect on the success and failure of both conflicts. Since Malaya Government had been co-operation with British Military Administration which lead to a successful in signing the peace between communist Thailand and Malaya in 1989. Although in Malaya television could not show the horrors of war during that time, but through the story from people to another we knew that the Malaya had establish a campaign to the citizen about the violence of the Communist towards Malaya citizen especially in isolated area. For example, Sir Robert Thompson offers six essential principles for how to succeed in counterinsurgency warfare. First, the government must have clear political aims, such as a free, stable, united country. Second, the government must function in accordance with the law. Only by doing so can the government preserve its legitimacy in the eyes of the people. Third, the government must have an overall plan for coordinating civil and military efforts. Fourth, the priority should be to defeat political subversion, not the guerillas themselves. Fifth, after military operations have been conducted in a specific area, civic action programs must be initiated. Finally, the government must have already secured its base areas first and by working outwards from their secure areas, government forces are more likely to have some morale-boosting successes early on. From this example we can surely know that this timeframe had led to the successful of the counterinsurgency in Malaya. While in Vietnam, historical tells us that United states eventually lost their war through lack of national support. This is because the Government Vietnam and United State army only focuses on the first layer in the devising their strategic Hamlet program as mention pervious point. The plan and strategy were dawn very neat with full of preparations, but when it comes to execution the whole thing became waste. The sensitivity to the citizen does not given priority as the program conduct. They execute the plan without taking into consideration to the population of the Vietnam’s. thus this kind of time frame are one of the reason that led to the failure of both conflicts.

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