138 Thermal History of the Universe
was stable in comparison with the age of the universe. At퐴=8 the nuclei^8 Band^8 Li
have lifetimes near one second which are not quite total bottlenecks. Because of these
gaps and bottlenecks and because^4 He is so strongly bound, nucleosynthesis essen-
tially stops after^4 He production. Only minute quantities of the stable nuclei^2 H,^3 He
and^7 Li remain with important relic abundances until today.
The fusion rate at energy퐸of two nuclei of charges푍 1 ,푍 2 is proportional to the
Gamow penetration factor
exp(
−
2 푍 1 푍 2
√
퐸
)
. (6.93)
Thus as the energy decreases, the fusion of nuclei other than the very lightest ones
becomes rapidly improbable.
Relic^4 He Abundance. The relic abundances of the light elements bear an important
testimony of the푛∕푝ratio at the time of the nucleosynthesis when the Universe was
only a few minutes old. In fact, this is the earliest testimony of the Big Bang we have.
Recombination occurred some 300 000 years later, when the stable ions captured all
the electrons to become neutral atoms. The CMB testimony is from that time. There is
also more recent information available in galaxy cluster observations from푧< 0 .25.
From the ratio in Equation (6.92) we obtain immediately the ratio of^4 He to^1 H:
푋 4 ≡
푁(^4 He)
푁(^1 H)=
푁n∕ 2
푁p−푁n≃^1
12
. (6.94)
The number of^4 He nuclei is clearly half the number of neutrons when the minute
amounts of^2 H,^3 He and^7 Li are neglected. The same number of protons as neutrons
go into^4 He, thus the excess number of protons becoming hydrogen is푁p−푁n.The
ratio of mass in^4 He to total mass in^1 Hand^4 He is
푌 4 ≡4 푋 4
1 + 4 푋 4
= 0. 2477 ± 0. 0001. (6.95)
from CMB measurements [8]. This is in some tension with the value 0.2551±0.0022
from [9]. This is a function of the ratio of baryons to photons
휂≡푁b
푁훾≃ 2. 75 × 10 −^8 훺bℎ^2 , (6.96)using푁훾from Table A.6.
Theheliummassabundance푌 4 depends sensitively on several parameters. If the
number of baryons increases,훺band휂also increase, and the entropy per baryon
decreases. Since the large entropy per baryon was the main obstacle to early deuteron
and helium production, the consequence is that helium production can start earlier.
But then the neutrons would have had less time to훽-decay, so the neutron/proton ratio
would be larger than^17. It follows that more helium will be produced:푌 4 increases.
The abundances of the light elements also depend on the neutron mean life휏nand
on the number of neutrino families퐹휈, both of which were poorly known until 1990.
Although휏nis now known to 1‰ [4], and퐹휈is known to be 3±4‰ 2, it may be