Big-bang nucJeosynthesis
23
abundances measured today differ significantly from those created in the first
three minutes. Because of this, the primordial abundances can only be inferred
from the observational data after corrections to allow for the effects of galactic
chemical evolution. The discussion of these is beyond the scope of this book.
Here we shall instead focus on the essential physics of the primordial processes.
Following Bemstein et al [6] and Sarkar [7], we shall present a semi-analytical
treatment that allows the 4He abundance to be calculated quite accurately. The
precise calculation of this and the other yields and, hence, of the present baryon
asymmetry requires a detailed numerical analysis which also will not be presented
here.
At sufficiently high temperatures (above a few MeV) neutrons and protons
are in kinetic and chemical equilibrium with, as follows from (1.129), a very
high value (O( 1 011» of the entropy per nucleon. During this era the equilibrium
nuclear abundances are quite negligible. The first stage of nucleosynthesis is the
freeze-out of the weak interaction processes
nVe ~ pene+ ~ PVe
n ~ pe-v e
( 1.136)
that previously kept neutrons and protons in equilibrium. Kinetic equilibrium
requires equality of the temperatures of the particles:
Tn = Tp = Te = Tv = T
(1.137)
and chemical eqUilibrium requires that the chemical potentials of the various
species satisfy
Iln - /l p = Ile- - /lv. = /lii. - Ile+'
( 1.138)
The total rate Anp for converting neutrons to protons via these processes is the
sum of the individual rates:
Anp = A(nve -+ pe-) + A(ne+ -+ pve) + A(n -+ pe-ve )
(1.139)
and the total rate Apn for the reactions that convert protons to neutrons is given by
detailed balance
Apn = Anpe-Am/T where ll.m = mn - mp = 1.293 MeV. ( 1.140)
(The difference arises because of the slightly different Boltzmann factors in the
neutron and proton equilibrium densities, see (4.19).) Let us denote the fractional
relative neutron abundance by Xn == "n/nN, where "n is the neutron number
density and "N is the total nucleon number density nN == "n + "p where np
is the proton number density. Then the fractional relative proton abundance is
Xp == np/nN = I - Xn. The evolution of Xn is determined by the balance
equation
Xn = Apn(l - Xn) - AnpXn.
(1.141)
23
abundances measured today differ significantly from those created in the first
three minutes. Because of this, the primordial abundances can only be inferred
from the observational data after corrections to allow for the effects of galactic
chemical evolution. The discussion of these is beyond the scope of this book.
Here we shall instead focus on the essential physics of the primordial processes.
Following Bemstein et al [6] and Sarkar [7], we shall present a semi-analytical
treatment that allows the 4He abundance to be calculated quite accurately. The
precise calculation of this and the other yields and, hence, of the present baryon
asymmetry requires a detailed numerical analysis which also will not be presented
here.
At sufficiently high temperatures (above a few MeV) neutrons and protons
are in kinetic and chemical equilibrium with, as follows from (1.129), a very
high value (O( 1 011» of the entropy per nucleon. During this era the equilibrium
nuclear abundances are quite negligible. The first stage of nucleosynthesis is the
freeze-out of the weak interaction processes
nVe ~ pene+ ~ PVe
n ~ pe-v e
( 1.136)
that previously kept neutrons and protons in equilibrium. Kinetic equilibrium
requires equality of the temperatures of the particles:
Tn = Tp = Te = Tv = T
(1.137)
and chemical eqUilibrium requires that the chemical potentials of the various
species satisfy
Iln - /l p = Ile- - /lv. = /lii. - Ile+'
( 1.138)
The total rate Anp for converting neutrons to protons via these processes is the
sum of the individual rates:
Anp = A(nve -+ pe-) + A(ne+ -+ pve) + A(n -+ pe-ve )
(1.139)
and the total rate Apn for the reactions that convert protons to neutrons is given by
detailed balance
Apn = Anpe-Am/T where ll.m = mn - mp = 1.293 MeV. ( 1.140)
(The difference arises because of the slightly different Boltzmann factors in the
neutron and proton equilibrium densities, see (4.19).) Let us denote the fractional
relative neutron abundance by Xn == "n/nN, where "n is the neutron number
density and "N is the total nucleon number density nN == "n + "p where np
is the proton number density. Then the fractional relative proton abundance is
Xp == np/nN = I - Xn. The evolution of Xn is determined by the balance
equation
Xn = Apn(l - Xn) - AnpXn.
(1.141)