Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
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326 9 Big Bang Nuc
l
eosynt
h
esis an
d
Re
l
ic Neutrinos
Table 9.1 The equation of state, energy density and scale parameter of a Universe
d
ominate
db
yra
d
iation, matter or vacuum ener
g
y.
D
ifferent era Equation of state Energy density Scale paramete
r
R
adiatio
n
p
=
ρ/
3
ρ
r
∝ R
−
4
∝
t
−
2
R
∝
t
1
/
2
Matte
r
p
≈
0
ρ
m
∝
R
−
3
∝
t
−
2
R
∝
t
2
/
3
Vacuum
p
=
−
ρρ
v
= cons
t
an
t
R
∝
e
xp
(
Λ
/
3
t
)
(
2
)
A
matter-dominated
U
nivers
e
.
A
t relativel
y
late times the Univers
e
g
radually cooled down, so non-relativistic matter eventually dominated the
energy density over radiation. An essentially pressureless gas corresponds t
o
the equation o
f
state
p
≈
0
(
i.e.,
w
≈
0)
2
,
imp
l
yin
g
ρ
m
∝
R
−
3
as o
n
ecan
s
ee from Eq.
(
9.6
)
. If the terms associated wit
h
ρ
r
,
ρ
v
a
nd
k
in Eq.
(
9.8
)
ar
e
omitte
d
,weo
b
tai
n
˙
R
∝
1
/R
1
/
2
.
Therefore
,
R
(
t
)
∝
t
2
/
3
a
n
d
H
=
2
/
(3
t
)
hold.
(
3
)
A
vacuum-dominated Universe
.
In quantum
fi
eld theor
y
the vacuum is
d
efined as the lowest energy state of a system and contains virtual particle
-
a
ntipartic
l
e pairs w
h
ic
h
spontaneous
l
yappearan
dd
isappear. T
h
e vacuum
ener
g
y
d
ensity
ρ
v
,
w
h
ic
h
acts as a cosmo
l
o
g
ica
l
constant
Λ
,
cou
ld d
ominat
e
theenergydensityoftheUniverseinanerawhen
R
(
t
)
is large enough. I
n
this case the vacuum state is described by the equation o
f
state
p
=
−
ρ
v
,
an
d
s
uch a ne
g
ative pressure is equivalent to a
g
ravitational repulsion. It is obviou
s
tha
t
ρ
v
i
s inde
p
endent of
R
(
t
)
, as a result o
f
w
=
−
1
. Neglecting the term
s
assoc
i
ated w
i
th
ρ
m
,
ρ
r
a
n
d
k
i
nEq.
(
9.8
)
,wesimplyobtain
H
=
˙
R
/R
=
Λ/
3
and thu
s
R
(
t
)
∝
exp
(
Λ/
3
t
)
. This result implies an exponential
expansion of the Universe. One may argue that the parameter
w
=
p/ρ
m
ig
ht
be
in
co
n
sta
n
ta
n
ddev
i
ate f
r
o
m
−
1 as a result o
f
the dynamically evolvin
g
vacuum energy or “dark energy”
(
Garnavich et al.
,
1998
;
Perlmutter
e
tal
.
,
1
999
;M
aor
e
ta
l
.
,
2002
)
.I
f
w
is a constant
,
t
h
e
b
est current measurement
f
or the equation o
f
state
y
ields
w
=
−
1
.
006
+
0
.
067
−
0
.
068
(
Komatsu
et al.
,
2009
)
.S
o
we simply assume that the vacuum energy is equivalent to a cosmological
co
n
sta
n
tw
i
th
w
=
−
1
.
O
ne may use the Friedmann equation in Eq.
(
9.8
)
to define the critica
l
energy density of the Universe which assures
k
=
0 to hold
:
ρ
c
≡
3
H
2
8
π
G
N
.
(
9.9
)
To
b
emoreexp
l
icit,
ρ
c
≈
1
.
05
×
10
−
5
h
2
G
e
V
cm
−
3
w
i
th
h
b
ein
g
t
h
e resca
l
e
d
H
ubble parameter. Toda
y
’s value o
f
the critical densit
y
is expected to b
e
ρ
c
≈
9
.
6
×
1
0
−
2
7
kg m
−
3
≈
5
.
4
×
1
0
−
6
G
e
V
cm
−
3
with the in
p
u
t
h
≈
0
.
7
2
.
2
I
n fact, the gas consisting of non-relativistic particles has the equation of state
p
=
(
2
ρ/
3
)
·
(
v
2
/
c
2
)(
Perkins, 2009
)
.Because
v
2
c
2
g
enerally holds
f
or cosmi
c
matter, the pressure that it exerts is very small and can be neglected.
9
.1 Neutrinos in t
h
eEar
l
y Universe 327
Th
ecosmo
l
o
g
ica
ld
ensity parameter
Ω
,w
h
ic
h
is a
l
so ca
ll
e
d
t
h
ec
l
osur
e
parameter, is defined as the ratio of the ener
g
y density to the critical density:
Ω
≡
ρ
ρ
c
=
1
+
k
H
2
R
2
,
(
9.10
)
where Eqs.
(
9.7
)
and
(
9.8
)
have been used. One can see that a flat Universe
wit
h
k
=
0musthav
e
Ω
=
1 for all values o
f
t
.When
Ω
> 1
,
k
=+
1holds
a
n
d
t
h
e
U
niverse is c
l
ose
d
.
Wh
en
Ω<
1
,
k
=
−
1h
o
ld
san
d
t
h
e
U
niverse i
s
open.
S
imilar de
fi
nition
s
Ω
m
≡
ρ
m
/ρ
c
,
Ω
r
≡
ρ
r
/
ρ
c
a
n
d
Ω
v
≡
ρ
v
/
ρ
c
lead to
the relationshi
p
Ω
=
Ω
m
+
Ω
r
+
Ω
v
.
(
9.11
)
A
t the present time
(
t
=
t
0
)
the density of the microwave photon radia-
tion is measure
dby
Ω
r
(
t
0
)
≈
4
.
84
×
10
−
5
a
n
d
t
h
us comp
l
ete
l
yne
gl
i
g
i
bl
ei
n
c
om
p
arison with that o
f
non-relativistic matte
r
Ω
m
(
t
0
)
≈
0
.
26, but the ma-
j
or contribution to
Ω
comes from the vacuum ter
m
Ω
v
(
t
0
)
≈
0
.
7
4i
f
k
=0
i
s assumed
(
Komats
u
et al
.
,
2009; Na
k
amura
et al.
, 2010
)
. Both luminou
s
bar
y
onic matter and dark matter contribute to
Ω
m
, and the latter actuall
y
d
ominates.
9.1.3 The A
g
e and Radius o
f
the Univers
e
L
et us estimate the a
g
eo
f
theUniversebymeanso
f
the Friedmann equation
.
N
ote tha
t
R
(
t
)
= R
(
t
0
)
/
(
1
+
z
)
holds, where
z
denotes the redshift. Eq.
(
9.9
)
m
eans
8
π
G
N
/
3=
H
2
(
t
)
/ρ
c
(
t
)=
H
2
0
HH
/ρ
c
(
t
0
)
,where
H
0
HH
is the
H
ubble co
n
sta
n
t
to
d
a
y
a
t
t
=
t
0
.ThenEq.
(
9.8
)
can be rewritten a
s
H
2
(
t
)=
H
2
0
HH
ρ
c
(
t
0
)
ρ
m
(
t
)
+
ρ
r
(
t
)
+
ρ
v
(
t
)
−
ρ
c
(
t
0
)
H
2
0
HH
·
k
R
2
(
t
)
=
H
2
0
HH
ρ
c
(
t
0
)
ρ
m
(
t
0
)(
1+
z
)
3
+
ρ
r
(
t
0
)(
1
+
z
)
4
+
ρ
v
(
t
0
)
−
ρ
c
(
t
0
)
H
2
0
H
H
·
k
R
2
(
t
0
)
(
1+
z
)
2
=
H
2
0
HH
Ω
m
(
t
0
)(
1+
z
)
3
+
Ω
r
(
t
0
)(
1
+
z
)
4
+
Ω
v
(
t
0
)
+[1
−
Ω
m
(
t
0
)
−
Ω
r
(
t
0
)
−
Ω
v
(
t
0
)] (
1+
z
)
2
,
(
9.12
)
where Table 9.1 and Eq.
(
9.10
)
have been used. Furthermore, one has
H
=
˙
R
R
=
−
1
1+
z
·
d
z
d
t
=
⇒
==
d
t
=
−
1
(
1+
z
)
H
d
z
,
(
9.13
)
with the help of Hubble’s law in Eq.
(
9.1
)
.Given
z
=
∞
at
t
= 0 and
z
=
0
at
t
=
t
0
,
it is possible to determine the age of the Universe
(
i.e., the valu
e
328 9 Big Bang Nuc
l
eosynt
h
esis an
d
Re
l
ic Neutrinos
of
t
0
)
by substituting Eq.
(
9.12
)
into Eq.
(
9.13
)
and then calculating the
i
nte
g
ration. For simplicity, here we assume the Universe to be flat wit
h
k
=
0
o
r
Ω
m
(
t
0
)+
Ω
r
(
t
0
)
+ Ω
v
(
t
0
)
= 1 and neglect the tiny Ω
r
(
t
0
)
term. The age
o
f
the
U
niverse turns out to be
t
0
=
1
H
0
HH
∞
0
d
z
(
1+z
)
Ω
m
(
t
0
)(
1
+
z
)
3
+
Ω
v
(
t
0
)
=
1
3
H
0
HH
Ω
v
(
t
0
)
l
n
1
+
Ω
v
(
t
0
)
1
−
Ω
v
(
t
0
)
,
(
9.14
)
w
h
ere
Ω
v
(
t
0
)
=
1
−
Ω
m
(
t
0
)
. Takin
g
Ω
m
(
t
0
)
≈
0
.
26,
Ω
v
(
t
0
)
≈
0
.
7
4an
d
h
≈
0
.
72
f
or example, we obtain
t
0
≈
1
/H
0
HH
≈
13
.
6Gy
r. This estimate is consisten
t
very well with the a
g
e of the Universe deduced from the study of its structure
f
ormation by means of the cosmic microwave background
(
CMB
)
and large
-
scale st
r
uctu
r
es:
t
0
=
13
.
69
±
0
.
1
3Gyr
(
Komatsu
et al.
,
2009
).
T
he radius of the observable Universe
,
defined a
s
D
H
,
is measured b
y
th
e
d
istance from the Earth to the optical horizon beyond which no light signal
s
c
ould reach the Earth at the present time
(
Perkins, 2009
)
. As the Universe
ex
p
ands and time evolves, this distance increases because more
p
arts of th
e
U
niverse come insi
d
et
h
e
h
orizon. Henc
e
D
H
>
ct
0
≈
4
.
2G
pc is expected
,
w
h
e
r
e
c
=
2
.
99792
4
5
8
×
1
0
8
m
s
−
1
i
s the speed o
f
li
g
ht which has been take
n
to be unity in the system of natural units. Note that the physical coordinate
d
istance
f
rom the Earth to any point o
f
the Universe at time
t
i
s expresse
d
as
D
(
t
)=
rR
(
t
)
with
r
bein
g
the comovin
g
coordinate distance.
S
ince bot
h
r an
d
R
(
t
)
are not directly measurable, we need to link them to the Hubbl
e
paramete
r
H
(
t
)
and the redshift
z
w
ith the help o
f
the Friedmann-Lemaˆıtre-
R
obertson-Walker metric. The latter is known as
d
s
2
=d
t
2
−
R
2
(
t
)
d
r
2
1
−
k
r
2
+
r
2
d
θ
2
+
s
in
2
θ
d
φ
2
.
(
9.15
)
Given a light signal to or from a distant object at fixed
(
θ, φ
)
,
d
s
2
=0h
o
lds
a
n
dthus
d
r
√
1
−
kr
2
=
d
t
R
(
t
)
=
−
d
z
HR
(
t
0
)
,
(
9.16
)
where Eq.
(
9.13
)
and
R
(
t
)=
R
(
t
0
)
/
(
1
+
z
)
have been used. We obtain
R
(
t
0
)
r
0
d
r
√
1
−
kr
2
=
−
0
z
d
z
H
=
I
(
z
)
H
0
HH
,
(
9.17
)
where Eq.
(
9.12
)
has been used, an
d
I
(
z
)
≡
z
0
Ω
m
(
t
0
)(
1+
z
)
3
+
Ω
r
(
t
0
)(
1
+
z
)
4
+
Ω
v
(
t
0
)
+[1
−
Ω
m
(
t
0
)
−
Ω
r
(
t
0
)
−
Ω
v
(
t
0
)] (
1+z
)
2
−
1
/
2
d
z
.
(
9.18
)
9
.1 Neutrinos in t
h
eEar
l
y Universe 329
T
h
esimp
l
est case is
k
= 0, correspondin
g
to a
fl
at Universe, in which
D
(
z
)=
r
R
(
t
0
)
=
I
(
z
)
/H
0
HH
as one can easily see from Eq.
(
9.17
)
. The horizon distanc
e
D
H
i
st
h
en o
b
taina
bl
e
b
y setting
z
=
∞
for the upper limit of the integration
i
nEq.
(
9.18
)
. Typically taking
Ω
r
(
t
0
)
=0,
Ω
m
(
t
0
)
=0
.
2
6
an
d
Ω
v
(
t
0
)
=
0
.
7
4
,
we find a numerical solutio
n
I
(
∞
)
≈
3
.
5
. The radius of the observable fla
t
U
niverse turns out to
b
e
D
H
∼
3
.
5
c/H
0
HH
≈
3
.
5
ct
0
≈
1
4
.
8Gp
c
≈
4
.
8
×
10
10
l
y
.
9.1.4 Rad
i
at
i
on
i
n the Earl
y
Un
i
verse
I
f
matter has been conserved in the Universe, its ener
g
y density is expecte
d
t
o vary as ρ
m
∝
R
−
3
.
In comparison, the energy density of radiation varies a
s
ρ
r
∝
T
4
∝
R
−
4
i
f it is in thermal equilibrium
(
see Table 9.1
)
.Theextr
a
R
−
1
in
ρ
r
,
as com
p
ared with
ρ
m
f
or non-relativistic matter, simpl
y
arises
f
rom
the redshift.
So
ρ
r
∝
R
−
4
applies not only to the CMB photons but also t
o
a
ny relativistic particles in the early Universe, provided they have a uni
f
orm
d
istribution on the same cosmological scale as the photons
(
Perkins, 2009
).
A
lthoug
h
ρ
m
ρ
r
h
o
ld
sto
d
ay, we
b
e
l
ieve t
h
a
t
ρ
r
ρ
m
an
d
ρ
r
ρ
v
mus
t
have bee
n
t
r
ue fo
r
s
m
a
ll
va
l
ues of
R
i
n the olden da
y
so
f
the Universe. I
n
that case the Friedmann equation in Eq.
(
9.8
)
can be simplified t
o
˙
R
R
2
=
8
π
G
N
3
ρ
r
,
(
9.19
)
b
ecause t
h
e term proportiona
l
to
k/R
2
is a
l
so ne
gl
i
g
i
bl
eascompare
d
wit
h
the term
p
ro
p
ortional t
o
ρ
r
∝ R
−
4
.
Therefore
,
˙
ρ
r
ρ
r
=
−
4
˙
R
R
=
−
8
3
6
π
G
N
ρ
r
,
(
9.20
)
l
eadin
g
to the ener
g
y density o
f
radiation
:
ρ
r
=
3
3
2
π
G
N
·
1
t
2
.
(
9.21
)
F
or a p
h
oton
g
as in t
h
erma
l
equi
l
i
b
rium, its ener
g
y
d
ensity is
g
iven
b
y
ρ
r
=
4
σ
S
B
T
4
=
g
γ
π
2
3
0
T
4
,
(
9.22
)
whe
r
e
σ
S
B
=
π
2
/
60
is the
S
te
f
an-Boltzmann constant an
d
g
γ
=
2
de
n
otes the
n
umber of spin substates of the photon. Combining Eqs.
(
9.21
)
and
(
9.22
)
a
llows us to obtain a relationshi
p
between the tem
p
erature of radiation and
thetimeo
f
expansion in the earl
y
Universe:
T
=
45
16
g
γ
π
3
G
N
1
/
4
1
√
t
.
(
9.23
)
330 9 Big Bang Nuc
l
eosynt
h
esis an
d
Re
l
ic Neutrinos
T
h
is resu
l
timp
l
ie
s
T
≈
1
.
3
1Me
V
/
√
t
≈
1
.
52
×
10
10
K
/
√
t
w
i
th
t
b
ein
g
i
n
s
econds. In view of
T
→
∞
as
t
→
0
,
we conclude that the Universe mus
t
h
ave starte
d
out as a
h
ot Big Bang.
Note that the spectrum o
f
blackbody photons o
f
ener
gy
E
=
p
i
s
g
iven
by
the Bose-Einstein distribution. The latter describes the number of
p
hotons
per unit vo
l
ume in t
h
e momentum interva
l
p
→
p
+d
p
d
.To
b
eexp
l
ici
t
3
,
n
γ
=
∞
0
g
γ
p
2
d
p
d
2
π
2
[
exp
(
E/T
)
−
1]
=
2
ζ
(
3
)
π
2
·
g
γ
2
T
3
≈
4
11
T
2
.
7
2
5K
3
cm
−
3
,
(
9.24
)
i
nwhic
h
ζ
(
3
)
≈
1
.
2
02 is a Riemann zeta function. Taking
T
=2
.
725 K as
today’s CMB temperature, one obtains the number density of relic photon
s
n
γ
≈
4
11
c
m
−
3
.
The total ener
g
y density o
f
photons inte
g
rated over their
blackbody spectrum turns out to b
e
ρ
r
=
∞
0
g
γ
Ep
2
d
p
d
2
π
2
[
exp
(
E
/T
)
− 1
]
=
g
γ
π
2
30
T
4
,
(
9.25
)
which is already given in Eq.
(
9.22
)
.Given
T
=
2
.
725 K to
d
a
y
at
t
=
t
0
,
the ener
g
y density of radiation i
s
ρ
r
(
t
0
)
≈
4
.
6
5
×
10
−
31
kg
m
−
3
≈
2
.
6
1
×
10
−
10
G
e
V
cm
−
3
.He
n
ce
Ω
r
(
t
0
)=
ρ
r
(
t
0
)
/ρ
c
≈
4
.
8
4
×
10
−
5
, about four order
s
o
f
ma
g
nitude smaller than
Ω
m
(
t
0
)
≈
0
.
26
.
Relativistic leptons and quarks can also contribute to the energy density
of radiation if they are sufficiently stable. A fermion gas obeys the Fermi
-
D
irac
d
istri
b
ution, an
d
t
h
e correspon
d
in
g
num
b
er
d
ensity in t
h
ere
l
ativistic
l
imit
(
i.e.,
T
m
an
d
E
=
p
2
+
m
2
→
p
wit
h
m
b
eing the fermion mas
s
a
n
d
p
b
eing the momentum
)
read
s
n
f
=
∞
0
g
f
p
2
d
p
d
2
π
2
[
exp
(
E
/T
)
+1
]
=
3
ζ
(
3
)
2
π
2
·
g
f
2
T
3
(
9.26
)
w
i
th
g
f
b
ein
g
the number o
f
spin substates o
f
the
f
ermion.
S
imilar t
o
ρ
r
in
E
q.
(
9.25
)
, the total energy density of relativistic fermions integrated over
the
ir F
e
rmi-Dir
ac d
i
st
ri
but
i
o
ni
s
ρ
f
=
∞
0
g
f
Ep
2
d
p
d
2
π
2
[
exp
(
E/
T
)
+1
]
=
7
g
f
π
2
24
0
T
4
.
(
9.27
)
F
or a mixture of extremely relativistic bosons and fermions, their total energ
y
d
ensity is expecte
d
to
be
3
F
our integrals are very use
f
ul in calculating the Bose-Einstein and Fermi-
Dirac
d
istri
b
utions:
7
∞
0
77
x
2
(
e
x
−
1)
−
1
d
x
=
2
ζ
(
3
),
7
∞
0
77
x
3
(e
x
−
1)
−
1
d
x
=
π
4
/
15,
7
∞
0
7
7
x
2
(
e
x
+
1
)
−
1
d
x
=3
ζ
(
3
)
/
2
an
d
7
∞
0
77
x
3
(
e
x
+
1
)
−
1
d
x
=
7
π
4
/
120
,
w
h
ere
ζ
(
3
)
≈
1
.
202 is a Riemann zeta function (Broadhurst, 2010).
9
.1 Neutrinos in t
h
eEar
l
y Universe 331
ρ
b
+f
=
b
∞
0
g
b
Ep
2
d
p
d
2
π
2
[
exp
(
E/T
)
−
1]
+
f
∞
0
g
f
Ep
2
d
p
d
2
π
2
[
exp
(
E/
T
)
+1
]
=
g
∗
π
2
30
T
4
,
(
9.28
)
where the effective number of degrees of freedo
m
g
∗
is given by
g
∗
=
b
g
b
+
7
8
f
g
f
(
9.29
)
with the summation over all types o
f
relativistic particles and antiparticle
s
which contribute to the ener
g
y density o
f
radiation in the early Universe
.
I
n the radiation-dominated e
p
och, the relation between T an
d
t
o
btaine
d
i
nEq.
(
9.23
)
should be modified by replacing
g
γ
w
i
th
g
∗
. As a result,
t
=
45
16
g
∗
π
3
1
/
2
M
Pl
MM
T
2
≈
0
.
301
√
g
√√
∗
·
M
Pl
MM
T
2
,
(
9.30
)
where
M
Pl
M
M
=
1
/
G
N
≈
1
.2
2
×
10
19
GeV is the Planck mass. To be s
p
ecific
,
tT
2
≈
2
.
42
/
√
g
√√
∗
s
Me
V
2
.
O
ne ma
y
also express the Hubble paramete
r
H
(
t
)
i
n terms of the tem
p
eratur
e
T
i
n the radiation-dominated epoch of the earl
y
U
niverse. Combining Eqs.
(
9.20
)
,
(
9.21
)
and
(
9.30
)
,weobtai
n
H
(
t
)
=
˙
R
R
=
1
2
t
≈
1
.
66
√
g
√√
∗
T
2
M
Pl
MM
.
(
9.31
)
When the temperature was hi
g
henou
g
h, all types o
f
elementary particles
a
nd their antiparticles would have been in the thermal bath of the earl
y
U
niverse. In t
h
is cas
e
g
∗
equals the number of all available spin and colo
r
s
ubstates of particles and antiparticles. Given the standard model
(
SM
)
o
f
electroweak and stron
g
interactions which contains one photon, ei
g
ht
g
luons,
t
h
ree massive gauge
b
osons, one Higgs
b
oson, six quar
k
s, six antiquar
k
s, t
h
re
e
ch
ar
g
e
dl
eptons, t
h
ree c
h
ar
g
e
d
anti
l
eptons, t
h
ree mass
l
ess neutrinos an
d
t
h
re
e
m
assless antineutrinos
4
,
the value o
f
g
∗
t
urns out to be
g
∗
=28
+7
/
8
×
9
0
=
106
.
75. T
h
is num
b
er wou
ld b
ecome sma
ll
er as t
h
eUniverseexpan
d
e
d
an
d
t
h
e
temperature fell. For example, the most massive particles
(
i.e., the top quark
,
the Higgs boson and th
e
W
±
a
nd
Z
0
bosons
)
would have been rapidly lost b
y
their decays
(
in less than 10
−
23
s
)
and not replenished once the temperature
T
i
s significantly below their masses
(
Perkins, 2009
)
.After
T
fe
ll
be
l
ow t
h
e
s
cale of quantum chromodynamics
Λ
QCD
∼
2
00 MeV, the remaining quarks
,
4
N
ote that each quark
(
or antiquark
)
has three color substates and each massless
neutrino
(
or antineutrino
)
has one spin substate. Although three known neutrinos
should be massive beyond the SM, their masses are so small that the coupling o
f
each neutrino to the additional spin substate is stron
g
ly suppressed and its e
ff
ect
on the value of
g
∗
i
s therefore negligible (Grupe
n
et a
l
., 2005)
.
332 9 Big Bang Nuc
l
eosynt
h
esis an
d
Re
l
ic Neutrinos
a
ntiquarks and
g
luons would no lon
g
er exist as separate components o
f
a
p
lasma but as color-neutral hadrons such as
p
ions and nucleons. However,
on
l
yprotonsan
d
neutrons cou
ld
survive since a
ll
t
h
eot
h
er
h
a
d
rons wou
ld b
e
too short-lived to exist be
y
ond the first few nanoseconds.
A
fte
r
T
2
0
Me
V
m
ost of the nucleons and antinucleons would have annihilated into radiation
a
nd the number of surviving nucleons was only about one billionth of th
e
n
umber o
f
photons.
Note that Eq.
(
9.30
)
, or equivalently
tT
2
≈
2
.
42
/
√
g
√√
∗
s
Me
V
2
,
can be used
to estimate what time a typical energy scale of particle physics corresponds
to.
G
ive
n
g
∗
=
106
.
7
5, t
h
eP
l
anc
k
sca
l
e
T
∼
10
1
9
G
eV and the Fermi scal
e
T
∼
1
0
2
G
eV correspond respectivel
y
to
t
∼
10
−
45
san
d
t
∼
1
0
−
11
safterthe
Bi
g
Ban
g
. The scale o
fg
rand uni
fi
ed theorie
s
T
∼
10
16
G
e
V
is associated with
t
∼
1
0
−
39
s
.
A
r
ou
n
dt
h
e sca
l
e
Λ
Q
C
D
, just be
f
ore the hadronization o
f
li
g
ht
q
uarks and gluons, one may take the photon, eight gluons, three light quark
s
(
u
,
d
,
s
)
, the electron, the muon, three neutrinos and their antiparticles to b
e
r
e
l
at
i
v
i
st
i
c
.In
t
hi
s case
g
∗
=18
+
7
/
8
×
50 = 61
.
75 holds, and thus
T
∼
Λ
Q
CD
c
orres
p
onds to
t
∼
7
.
7
×
10
−
6
s after the Big Bang
.
9.1.5 Neutrino Decouplin
g
When the temperature went down to a few MeV, the only surviving rel
-
a
tivistic partic
l
es in t
h
eear
ly
Universe were p
h
otons, e
l
ectrons, positrons
,
n
eutrinos and antineutrinos. In this case the effective number of de
g
rees of
f
reedom
g
∗
t
oo
k
its va
l
ue
g
∗
=
2+7
/
8
×
10
=
10
.
7
5. T
h
ere
l
evant p
h
otons,
l
ep
-
tons an
d
anti
l
eptons s
h
ou
ld h
ave
b
een present in compara
bl
enum
b
ers t
h
an
ks
to the e
q
uilibrium reaction
s
γ
+
γ
e
+
+
e
−
ν
α
+
ν
α
(
fo
r
α
=
e, μ,
τ
)
.
T
he
e
+
+
e
−
ν
α
+
ν
α
scattering is a wea
k
neutra
l
-current process w
h
ose
t
h
erma
ll
y-avera
g
e
d
cross section is rou
ghl
y
σ
e
±
v
e
±
∼
G
2
F
T
2
,
in w
h
ic
h
v
e
±
d
enotes the relative velocit
y
of electrons and positrons. The collision rat
e
fo
r
t
hi
s
r
eact
i
o
ni
s
Γ
=
n
e
±
σ
e
±
v
e
±
w
i
th
n
e
±
being the number density of
electrons or positrons.
G
iven
n
e
±
∼
T
3
at
T
as one can see from Eq.
(
9.26
),
the collision rate turns out to be
Γ
∼
G
2
F
T
5
.
In com
p
arison, the ex
p
ansion
rate o
f
the Universe in the radiation-dominated epoch is
H
∼
√
g
√√
∗
T
2
/M
Pl
M
M
as
s
hown in Eq.
(
9.31
)
. Hence neutrinos and antineutrinos would become out
of e
q
uilibrium and decou
p
ledfromthethermalbathassoona
s
Γ
<
H
f
or
s
u
ffi
ciently small
T
.
Arou
g
h estimate yields the “freeze-out” temperature
T
fr
T
T
∼
√
g
√√
∗
G
2
F
M
Pl
MM
1
/
3
∼
1MeV
(
9.32
)
f
or neutrinos to
g
oouto
f
equilibrium. We there
f
ore expect that neutrino
s
a
nd antineutrinos froze out fo
r
T<T
fr
TT
a
nd then evolved in a wa
y
indepen-
d
ent of other particles and radiatio
n
5
.
In ot
h
er wor
d
s
,
t
h
eUniverse
b
ecame
5
D
i
ff
erent
f
rom muon and tau neutrinos
,
the electron neutrinos and their an-
tiparticles could interact with the nucleons via the charged-current weak processes
9
.1 Neutrinos in t
h
eEar
l
y Universe 333
transparent to neutrinos an
d
antineutrinos, w
h
ose momenta wou
ld
simp
ly
redshift with the cosmic ex
p
ansion. So the effective neutrino tem
p
erature fel
l
as
T
∼
1
/R
,
just like the temperature of photons. The number density of the
d
ecoup
l
e
d
neutrinos wou
ld
continue to
d
ecrease in proportion to
1
/R
3
a
n
d
c
ontribute to
g
∗
,
if the
y
were stable and relativistic.
S
oon after the neutrino decoupling
(
i.e., whe
n
T
m
e
)
, electrons and
positions
b
e
g
an to anni
h
i
l
ate vi
a
e
+
+
e
−
→
γ
+
γ
. The ener
g
y released
f
rom
this annihilation process would heat up the photon gas relative to the neu-
trino gas, because the latter was already decoupled. The enhancement of th
e
photon temperature can be calculated b
y
means o
f
the entrop
y
conservation
.
The entropy per unit volume of the particle gas is given b
y
s
=
7
d
Q/
T
w
i
th
Q
b
ein
g
the ener
g
y content per unit volume o
f
photons, electrons and
p
os
i
trons at tem
p
eratur
e
T
. Considering Eq.
(
9.28
)
and takin
g
Q
=
ρ
b+f
wit
h
g
∗
=
2+
7
/
8
× 4=11
/
2
,
we hav
e
s
=
2
g
∗
π
2
1
5
T
2
d
T
=
2
g
∗
π
2
45
T
3
=
11
4
·
2
g
γ
π
2
45
T
3
.
(
9.33
)
A
fter the annihilation of electrons and positrons the photons must have at-
taine
d
a temperature
T
γ
TT
w
it
h
t
h
e correspon
d
in
g
entropy
d
ensity
s
γ
=
2
g
γ
π
2
1
5
T
2
γ
TT
d
T
γ
TT
=
2
g
γ
π
2
45
T
3
γ
TT
.
(
9.34
)
Because the expansion of the Universe keeps adiabatic or isentropic,
s
γ
=
s
h
olds.
A
s a result
,
T
γ
T
T
=
11
4
1
/
3
T
.
(
9.35
)
In comparison, t
h
ere
l
ic neutrinos receive
d
no
b
oost an
d
t
h
eir temperatur
e
sh
ou
ld j
ust
be
T
ν
TT
=
T
.T
h
ere
l
ations
h
ip
b
etwee
n
T
ν
TT
a
n
d
T
γ
TT
tu
rn
souttobe
T
ν
T
T
=
4
11
1
/
3
T
γ
T
T
.
(
9.36
)
F
or p
h
otons an
d
re
l
ic neutrinos in t
h
is epoc
h
,t
h
e tota
l
re
l
ativistic energy
d
ensity is
g
iven
by
ρ
γ
+
ν
=
ρ
γ
+
ρ
ν
=
g
γ
π
2
30
T
4
γ
TT
+
7
8
·
g
ν
π
2
30
T
4
ν
TT
=
g
∗
π
2
30
T
4
γ
TT
,
(
9.37
)
where the e
ff
ective number o
f
de
g
rees o
ff
reedo
m
g
∗
i
sdefi
n
ed as
ν
e
+
n
e
−
+
p
, ν
e
+
p
e
+
+
n
a
nd
ν
e
+
e
−
+
p
n. The point wher
e
Γ
∼
H
h
el
d
for these reactions determines the freeze-out temperature, which is approximately
the same as that given in Eq.
(
9.32
)
. So all neutrino flavors were decoupled from
other particles below
T
fr
T
T
∼
1
MeV.
334 9 Big Bang Nuc
l
eosynt
h
esis an
d
Re
l
ic Neutrinos
g
∗
≡
g
γ
+
7
8
g
ν
T
ν
TT
T
γ
TT
4
=2
+
6
×
7
8
4
11
4
/
3
≈
3
.
36
,
(
9.38
)
a
pp
l
yin
g
to t
h
ere
g
ion wit
h
T
1
MeV
(
Kolb and Turner, 1990
)
. The total
entropy density of photons and relic neutrinos is therefore given b
y
s
γ
+
ν
=
s
γ
+
s
ν
=
2
g
γ
π
2
4
5
T
3
γ
TT
+
7
8
·
2
g
ν
π
2
4
5
T
3
ν
TT
=
2
π
2
45
1
+
7
8
·
g
ν
g
γ
T
ν
TT
T
γ
TT
3
g
γ
T
3
γ
T
T
=
2
π
2
45
·
π
2
ζ
(
3
)
1+
7
8
×
6
2
×
4
11
n
γ
≈
7
.
0
4
n
γ
,
(
9.39
)
where Eqs.
(
9.24
)
and
(
9.36
)
have been used. Although the neutrinos and
p
h
otons
h
ave
h
a
d
no more interactions since t
h
eneutrino
d
ecoup
l
in
g
,t
h
ey
h
ave been suffering from the same redshift as the Universe expands and cools
d
own. Hence t
h
eir re
l
ative num
b
er
d
ensities to
d
ay s
h
ou
ld b
et
h
esameas
those at the time of neutrino decoupling. With the help of Eqs.
(
9.24
)
and
(
9.26
)
, the number density of relic neutrinos and antineutrinos of all three
fl
avors is expected to b
e
n
ν
=
3
ζ
(
3
)
2
π
2
·
g
ν
2
T
3
ν
TT
=
3
4
·
g
ν
g
γ
T
ν
TT
T
γ
TT
3
n
γ
=
9
11
n
γ
≈
336
T
γ
TT
2
.
725 K
3
c
m
−
3
.
(
9.40
)
Ta
k
in
g
T
γ
T
T
=
2
.
7
25 K
f
or the
C
MB radiation toda
y
, we immediatel
y
obtain
T
ν
TT
≈
1
.
945 K an
d
n
ν
≈
336 c
m
−
3
f
or the
C
ν
B.
S
o far we have assumed that neutrinos were fully decoupled at
T
fr
T
T
∼
1
MeV and thus did not share an
y
entrop
yf
rom the
e
±
a
nnihil
at
i
o
n
after
T
was below m
e
.
In practice, such an instantaneous neutrino decouplin
g
a
pproximation s
h
ou
ld b
es
l
i
gh
t
l
y correcte
db
ecause neutrinos cou
ld
sti
ll h
ave
s
li
g
ht interactions with the electroma
g
netic plasma at
T<
1
MeV and hence
received a small portion of the entropy from the
e
±
a
nnihilation
(
Dicu
s
e
tal
.
,
1
982
)
. Those more energetic neutrinos may be more heated, because the wea
k
i
nteractions in the relevant ran
g
eofener
g
ies
g
et stron
g
er with risin
g
ener
g
ies
.
This phenomenon results in a momentum-dependent
(
non-thermal
)
distortio
n
i
n the neutrino distribution from the thermal equilibrium case
(
Mangano
et al.
,
2002
)
. The small effect of a non-instantaneous neutrino decoupling
,
together with a much smaller effect on
T
γ
TT
/T
ν
TT
induced by finite-temperatur
e
q
uantum e
l
ectro
d
ynamics corrections to t
h
ee
l
ectroma
g
netic p
l
asma, can
be
taken into account by defining an effective number of neutrino species in the
effective number of degrees of freedom
g
∗
given in Eq.
(
9.38
)
. Namely,
g
∗
=
2
1+
21
8
4
11
4
/
3
=
⇒
==
g
∗
=2
1
+
7
8
4
11
4
/
3
N
eff
ν
NN
,
(
9.41
)
9.2 Big Bang Nuc
l
eosynt
h
esis 33
5
whe
r
e
N
eff
ν
N
N
=3
has been taken in the limit o
f
an instantaneous neutrin
o
d
ecouplin
g
.
A
careful numerical analysis shows tha
t
N
eff
ν
N
N
≈
3
.
04
(
Seljak and
Za
ld
arriaga, 1996; Mangano
e
ta
l.
,
2002
).
Note t
h
at we
h
ave assume
d
neutrinos to
b
eextreme
ly
re
l
ativistic in t
he
a
bove discussions by ignoring their rest masses. Current neutrino oscillatio
n
d
ata in
d
icate t
h
at at
l
east two neutrinos s
h
ou
ld b
e massive.
Wh
en
T
ν
T
T
was
c
omparable with a neutrino mass, this kind o
f
neutrinos would no more be
relativistic. Because bot
h
T
ν
TT
an
d
T
γ
TT
f
ell as
1
/
R
a
fter neutrinos were decou
-
pled from the thermal bath, their relationship obtained in Eq.
(
9.36
)
in th
e
relativistic limit should essentiall
y
keep valid even i
f
one or all o
f
three neu
-
trino s
p
ecies became non-relativistic. For a similar reason we ex
p
ect tha
t
the relationship between the number densities o
f
relic neutrinos and photon
s
obtained in Eq.
(
9.40
)
should essentially keep unchanged even today a
t
t
=
t
0
.
Note also that we have neglected the effect of neutrino oscillations on th
e
process o
f
neutrino decouplin
g
in the above discussions. This e
ff
ect can b
e
taken into account by solvin
g
the momentum-dependent kinetic equations
f
or
the neutrino spectra
(
Dolgov
e
ta
l
.
,
2002; Hannesta
d
, 2002; Mangano
e
ta
l.
,
2
005
)
. It is found that neutrino oscillations in the early Universe should not
h
ave an appreciable impact on the neutrino ener
g
y density, so their effect o
n
the BBN should likewise be inappreciable
(
Mangano et a
l
.
,
2005
)
.
9
.2 Bi
g
Ban
g
Nucleosynthesis
O
ne o
f
the major achievements in the standard Bi
g
Ban
g
cosmolo
g
yisth
e
BBN theor
y
which success
f
ull
y
describes how protons and neutrons coul
d
f
use together and form the light elements
(
e.g., D,
3
H,
3
H
e
,
4
H
eand
7
L
i
)
in
the first few minutes after the Bi
g
Ban
g
. All of the heavier elements wer
e
f
ormedmuchlater
f
rom the s
y
nthesis o
f
nuclei in stars. We shall outline th
e
BBN timeline in this section and em
p
hasize that the
p
redictions of the BBN
theory
f
or the abundances o
f
the li
g
ht elements are in
g
ood a
g
reement wit
h
the primordial abundances of those elements observed toda
y
in the cosmos.
I
ts
h
ou
ld b
e note
d
t
h
at t
h
e
l
ig
h
te
l
ement a
b
un
d
ances o
b
serve
d
to
d
ay ar
e
n
ot a
ll
“primor
d
ia
l”
,
b
ecause our epoc
h
is certain
ly
muc
hl
ater t
h
an t
h
e BBN
era and the stellar nucleosynthesis has already commenced for a lon
g
time
.
The ejected remains of this stellar processing may alter the light elemen
t
a
bundances
f
rom their primordial values and produce some heav
y
element
s
s
uch as C, N, O and Fe
(
“metals”
)(
Nakamur
a
e
tal
.
, 2010
)
.Inthiscas
e
one should look for the astrophysical sites with low metal abundances s
o
a
s to measure t
h
e
l
i
gh
te
l
ement a
b
un
d
ances w
h
ic
h
are muc
h
c
l
oser to t
he
primordial ones. For all the light elements, the precision with which their
primordial abundances can be in
f
erred is mainly limited by systematic error
s
of t
h
e
m
easu
r
e
m
e
n
ts.