94
Baryogenesis
assuming that the energy density p = (1r 2 /30)g.,TT 4 is dominated by relativistic
particles. Here
( Ti)4 7
(Ti)4
g.,T = L gi T + 8" ~ gi T
(4.22)
bosons
ferrruons
satisfies g •• T = g",S.T = N. (with N. defined in (1.104» when all particle species
i are at the same temperature Ti = T. (See section 5.1.) The thermal average
(Uann V) '" m- 2
(4.23)
-
1r
and, at T = Tf ~ 20 MeV, the annihilation rate falls below the expansion rate,
nucleons and antinucleons are so dilute that they cannot annihilate further and
their number densities become frozen at
2
nN = nW = ~ 2 ( mN )3/2 e-mN / T1
(4.24)
ny
ny
{(3) 21fTf
~ 10- 18
(4.25)
using (4.6). This is far smaller than the value (4.12) which derives from the
measured primordial abundances of the light nuclei. Thus, the assumed zero
initial asymmetry is inconsistent with the nucleosynthesis data.
Of course, statistical fluctuations can generate a non-zero initial asymmetry.
At present our galaxy contains about 10 79 photons and 10 69 nucleons. When
T ;:: I GeV, however, the comoving volume containing our galaxy contained
about 10 79 baryons and antibaryons. Thus. statistical fluctuations might generate
an asymmetry
NB - NJj '" .IN
(4.26)
so that, instead of (4.19), we have
1
nN - nW '" - n N '" 1O- 39 . 5 n
(4.27)
.IN
N
which again is far too small to explain the nucleosynthesis data.
The conclusion is that the initial baryon asymmetry must be non-zero to
explain the size of the asymmetry we observe today. Of course, the required value
may be input by hand as an initial condition but aesthetically this is unattractive.
The consensus is that the asymmetry derives from new physics in the early
universe. We turn next to the three necessary conditions for baryogenesis, first
derived by Sakharov [6].
4.2 Conditions for baryogenesis
If we start from a universe with a net baryon number B of zero and evolve to
one with a non-zero value, it is clear that baryon number is not conserved. Thus
Baryogenesis
assuming that the energy density p = (1r 2 /30)g.,TT 4 is dominated by relativistic
particles. Here
( Ti)4 7
(Ti)4
g.,T = L gi T + 8" ~ gi T
(4.22)
bosons
ferrruons
satisfies g •• T = g",S.T = N. (with N. defined in (1.104» when all particle species
i are at the same temperature Ti = T. (See section 5.1.) The thermal average
(Uann V) '" m- 2
(4.23)
-
1r
and, at T = Tf ~ 20 MeV, the annihilation rate falls below the expansion rate,
nucleons and antinucleons are so dilute that they cannot annihilate further and
their number densities become frozen at
2
nN = nW = ~ 2 ( mN )3/2 e-mN / T1
(4.24)
ny
ny
{(3) 21fTf
~ 10- 18
(4.25)
using (4.6). This is far smaller than the value (4.12) which derives from the
measured primordial abundances of the light nuclei. Thus, the assumed zero
initial asymmetry is inconsistent with the nucleosynthesis data.
Of course, statistical fluctuations can generate a non-zero initial asymmetry.
At present our galaxy contains about 10 79 photons and 10 69 nucleons. When
T ;:: I GeV, however, the comoving volume containing our galaxy contained
about 10 79 baryons and antibaryons. Thus. statistical fluctuations might generate
an asymmetry
NB - NJj '" .IN
(4.26)
so that, instead of (4.19), we have
1
nN - nW '" - n N '" 1O- 39 . 5 n
(4.27)
.IN
N
which again is far too small to explain the nucleosynthesis data.
The conclusion is that the initial baryon asymmetry must be non-zero to
explain the size of the asymmetry we observe today. Of course, the required value
may be input by hand as an initial condition but aesthetically this is unattractive.
The consensus is that the asymmetry derives from new physics in the early
universe. We turn next to the three necessary conditions for baryogenesis, first
derived by Sakharov [6].
4.2 Conditions for baryogenesis
If we start from a universe with a net baryon number B of zero and evolve to
one with a non-zero value, it is clear that baryon number is not conserved. Thus