142 Thermal History of the Universe
6.6 Baryosynthesis and Antimatter Generation
In Equation (6.96) we noted that the ratio휂of the baryon number density푁Bto the
photon number density푁훾is very small. We shall see later that it is even two orders
of magnitude smaller.
The Ratio of Baryons to Photons. Before the baryons were formed (at about
200MeV), the conserved baryon number퐵was carried by quarks. Thus the total value
of퐵carried by all protons and neutrons today should equal the total value carried by
all quarks. It is very surprising then that휂should be so small today, because when
the quarks, leptons and photons were in thermal equilibrium there should have been
equal numbers of quarks and anti-quarks, leptons and anti-leptons, and the value of
퐵was equal to the total leptonic number퐿,
푁B=푁B=푁L≈푁훾 (6.100)because of the푈( 1 )퐵−퐿symmetry.
When the baryons and anti-baryons became nonrelativistic, the numbers of
baryons and anti-baryons were reduced by annihilation, so푁Bdecreased rapidly by
the exponential factor in the Maxwell–Boltzmann distribution [Equation (6.29)]. The
number density of photons푁훾is given by Equation (6.12) at all temperatures. Thus
the temperature dependence of휂should be
휂=
푁B
푁훾
=
√
2 휋
4. 808
(푚
푁
kT) 3 ∕ 2
e−푚푁∕kT. (6.101)When the annihilation rate became slower than the expansion rate, the value of휂
was frozen, and thus comparable to its value today. The freeze-out occurs at about
20MeV, when휂has reached the value
휂≃ 6. 8 × 10 −^19. (6.102)Butthisisafactor9× 108 too small! Thus something must be seriously wrong with
our initial condition [Equation (6.100)].
Baryon–Anti-baryon Asymmetry. The other surprising thing is that no anti-baryons
seem to have survived. At temperatures above 200MeV, quarks and anti-quarks were
in thermal equilibrium with the photons because of reactions such as Equations
(6.30)–(6.32), as well as
훾+훾↔q+q. (6.103)These reactions conserve baryon number, so every quark produced or annihilated
is accompanied by one anti-quark produced or annihilated. Since all quarks and
anti-quarks did not have time to annihilate one another, it would be reasonable to
expect equal number densities of baryons or anti-baryons today.
But we know that the Earth is only matter and not antimatter. The solar wind does
not produce annihilations with the Earth or with the other planets, so we know that