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