Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
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336 9 Big Bang Nuc
l
eosynt
h
esis an
d
Re
l
ic Neutrinos
9.2.1 The Neutron-to-proton Rat
io
In the radiation-dominated epoch of the early Universe, the synthesis of th
e
l
i
g
ht elements took place at temperatures varyin
g
rou
g
hly
f
rom about 1 Me
V
to below 0.1 MeV. During this BBN period the tim
e
t
and the tem
p
eratur
e
T
a
re re
l
ate
d
to eac
h
ot
h
er t
h
roug
h
t
h
ere
l
ations
h
ip
t
T
2
≈
2
.
42
/
√
g
√√
∗
sM
e
V
2
,
where the e
ff
ective number o
f
de
g
rees o
ff
reedo
m
g
∗
c
hanges from 10.75
(
fo
r
relativisti
c
γ
,
e
−
,
ν
e
,
ν
μ
,
ν
τ
a
nd their antiparticles
)
to 3.36
(
as
T
d
ro
pp
ed
below the elect
r
o
nm
ass
m
e
≈
0
.
511 MeV
)
. At a higher temperature such
as
T
∼
200 MeV, t
h
e
gl
uons, quar
k
san
d
antiquar
k
swere
b
oun
d
into nuc
l
e-
ons and antinucleons. The latter could remain in thermal e
q
uilibrium as the
temperature
d
roppe
d
t
o
T
∼
5
0MeV,
b
ut t
h
eir num
b
er
d
ensities were expo-
n
entially suppressed by a factor exp
(
−
m
N
/
T
)
wit
h
m
N
≈
0
.
94 GeV
(
Grupe
n
et al
.
, 2005
)
. At temperatures below about 20 MeV most of the nucleons an
d
a
ntinuc
l
eons s
h
ou
ld h
ave anni
h
i
l
ate
d
an
d
t
h
us t
h
e resu
l
tin
g
nuc
l
eon
d
ensit
y
was unable to make a si
g
ni
fi
cant contribution to the total ener
g
y density o
f
the Universe. Since there are not any imaginable baryon-number-violatin
g
processes t
h
at wou
ld h
ave occurre
d
at temperatures near an
d
in t
h
e BBN
era, the bar
y
on number conservation assures the sum of the number densities
of non-relativistic protons and neutrons to be constant in a comoving volume
a
n
d
c
h
an
g
eas
n
p
+
n
n
∼
T
3
∼
R
−
3
in a p
hy
sica
l
coor
d
inate s
y
stem
.
B
efore the decouplin
g
or freeze-out of electron neutrinos and electro
n
a
ntineutrinos, the following charged-current weak interactions allow the bi-
l
ateral conversion o
f
neutrons and protons:
n
+
ν
e
p
+
e
−
,p
+
ν
e
n
+
e
+
,n
p
+
e
−
+
ν
e
.
(
9.42
)
Because of
T
m
N
≈
0
.
9
4 GeV in the epoch under discussion, the nucleons
c
an be treated as essentially at rest. The initial and
fi
nal lepton ener
g
ies i
n
each of the above
p
rocesses are then related to each other vi
a
E
e
−
−
E
ν
e
=
Q
(
fo
r
n
+
ν
e
p
+
e
−
)
,
E
ν
e
−
E
e
+
=
Q
(
fo
r
p
+ν
e
n
+
e
+
)
o
r
E
e
−
+
E
ν
e
=
Q
(
fo
r
n
p
+
e
−
+
ν
e
)
,where
Q
=
m
n
−
m
p
=
1
.
293 MeV. The reaction
s
i
nEq.
(
9.42
)
were in thermal equilibrium fo
r
T
T
fr
TT
, because they coul
d
proceed faster than the expansion rate of the Universe
(
i.e.,
Γ>H
)
.Inthi
s
c
ase the ratio o
f
the neutron and proton number densities is
g
iven by
6
n
n
n
p
=
m
n
m
p
3
/
2
e
−
(
m
n
−
m
p
)
/
T
≈
e
−
Q/
T
.
(
9.43
)
Tentatively ignoring the effect of neutron decays and taking
T
fr
TT
≈
0
.
7
2
M
e
V
a
s the freeze-out temperature of neutrinos from a more careful analysis
(
Gru
-
pen
et al
.
, 2005
)
, one may obtai
n
n
n
/n
p
≈
1
/
6at
T
fr
TT
c
orresponding to
6
I
n the non-relativistic limit the number density of either bosons or fermions
can
b
e
d
escri
b
e
db
y
n
=
g
(
mT
)
3
/
2
/
(2
π
)
3
/
2
exp
(
−
m/
T
)
, where
g
i
st
h
e
n
u
m
be
r
of
internal de
g
rees o
ff
reedom
f
or the particle in question an
d
m
d
enotes its rest mass.
This result is valid in the assumption that the chemical potentials are negligible.
9.2 Big Bang Nuc
l
eosynt
h
esis 337
t
≈
1
.
4
sa
f
ter the Bi
g
Ban
g
.Belowthe
f
reeze-out temperature the neu-
trons were free to deca
y
vi
a
n
→
p
+
e
−
+
ν
e
.
This
p
rocess lasted a fe
w
m
inutes before the neutrons were bound up into deuterium and started the
c
hain o
f
nucleosynthesis reactions. Durin
g
this period the ratio
n
n
/n
p
r
eads
n
n
(
t
)
n
p
(
t
)
=
e
−
Q/T
fr
TT
e
−
t
/τ
n
1
+e
−
Q/T
fr
TT
1
−
e
−
t
/
τ
n
≈
e
−
Q/T
fr
T
T
e
−
t
/τ
n
,
(
9.44
)
whe
r
e
t
means the time be
g
innin
gf
rom the
f
reeze-out point to the start o
f
the nucleos
y
nthesis, and
τ
n
ττ
=(
88
5
.
7
±
0
.
8
)
s is the mean lifetime of neutrons
.
At a t
im
e
t
≈
160 s
(
i.e.,
t
≈
158
.
6
san
d
T ≈
0
.
09
Me
V
for
g
∗
≈
3
.
36
),
t
h
e neutron-to-proton ratio
d
roppe
d
t
o
n
n
/
n
p
≈
0
.
14. Fr
o
m
he
r
eo
n
out the
U
niverse should be cold enough that the BBN could start from scratch.
9.2.2 Synthesis o
f
the Light Nucle
i
The nucleosynthesis chain be
g
an with the formation of deuterium in the re
-
a
ctio
n
n
+
p
→
D+
γ wit
h
t
h
e
b
in
d
ing energy
E
b
in
d
≈
2
.
22 Me
V
.T
h
is i
s
a
ne
l
ectroma
g
netic process w
h
ic
hh
as a cross section muc
hl
ar
g
er t
h
an t
h
ose
of the weak processes listed in Eq.
(
9.42
)
, and thus it could stay in therma
l
equilibrium for quite a while. In other words, the deuterium would be bro-
ken apart as soon as it was produced i
f
there were man
y
photons and their
energies were greater than
E
b
in
d
.
Such a photodissociation process actuall
y
d
elayed the production of deuterium after the temperatur
e
T
d
roppe
db
e-
low
E
b
in
d
.
De
fi
nin
g
the ratio o
f
the baryon and photon number densities a
s
η
≡
n
B
/
n
γ
,onema
y
use the quantit
y1
/
η
e
xp
(
−
E
bind
/
T
)
to approximately
estimate the number of photons per baryon
(
i.e., per nucleon
)
above the
d
euterium photodissociation threshold ener
g
y. Note that
η
i
s the onl
yf
re
e
parameter in the BBN theory. Give
n
η
≈
6
×
10
−
10
,thenumberof
p
hoton
s
wit
h
ener
g
ies a
b
ove
E
b
in
d
i
s
f
ound to be rou
g
hly equal to that o
f
baryons a
t
T
∼
0
.
1MeVo
r
t
∼
132 s.
S
omewhat below this tem
p
erature the deuteriu
m
c
ould begin to form without being immediately photodissociated again.
O
ver the next
f
ew minutes, almost all the neutrons that had survived
s
ince the BBN started were processed into the most stable li
g
ht elemen
t
4
He
via a number of dominant two-body processes such as D +
n
→
3
H
+ γ
,
D
+
D
→
3
H
+
p
,D
+
p
→
3
H
e
+
γ
,D+D
→
3
H
e+
n
,an
d
D
+
D
→
4
H
e
+
γ,
3
H+
p
→
4
He
+
γ
,
D+
3
H
→
4
H
e
+
n,
D+
3
He
→
4
H
e+
p
.
(
9.45
)
Much heavier nuclei could not
f
orm in any si
g
ni
fi
cant quantity both because
of the absence of stable nuclei with the mass numbe
r
A
=
5
,
6 or 8 and due
to the large Coulomb barriers for some relevant reactions
(
Weinberg, 2008;
P
erkins, 2009; Nakamur
a
et al.
,
2010
)
. As an example
,
7
L
i
a
n
d
7
Be cou
l
dbe
produced through the processe
s
338 9 Big Bang Nuc
l
eosynt
h
esis an
d
Re
l
ic Neutrinos
3
H
+
4
H
e
→
7
L
i+
γ,
3
H
e+
4
He
→
7
B
e+
γ,
7
B
e
+
e
−
→
7
L
i
+
ν
e
,
7
Be
+
n
→
7
L
i
+
p
.
(
9.46
)
O
ne may there
f
ore estimate the primordial mass
f
raction o
f
4
H
e
(
i.e., th
e
r
at
i
ooft
h
e
m
asses of a
l
l
4
H
e elements to those of all nuclei
)
, conventionally
referred to as
Y
p
YY
=
m
He
n
He
/
m
N
n
p
+
n
n
,
by assuming that all of th
e
n
eutrons en
d
e
d
up in
4
He. Name
l
y, we
h
av
e
n
H
e
≈
n
n
/
2a
n
d
m
H
e
≈
4
m
N
to
g
ether wit
h
m
N
≈
m
p
≈
m
n
≈
0
.
9
4
G
eV. In this a
pp
roximation,
Y
p
YY
≈
2
n
n
n
p
+
n
n
≈
0
.
2
4
6
,
(
9.47
)
w
h
e
r
e
n
n
/
n
p
≈
0
.
1
4at
T
≈
0
.
0
9MeVo
r
t
≈
160 s has t
y
picall
y
been input
.
It should be
p
ointed out that the observe
d
4
H
e abundance in the cosmo
s
a
s
g
iven below is consistent with the above prediction
f
rom the BBN theor
y
a
nd far
g
reater than the one which could have been produced from hydro
g
en
burning in main-sequence stars
(
Perkins, 2009
)
.Henc
e
Y
p
YY
i
stru
l
y a measure
o
f
the primordia
l
4
He abu
n
da
n
ce.
A
semi-
q
uantitative but more accurate ex
p
ression of th
e
4
He mass fractio
n
i
sgivenby
(
Cirelli
e
ta
l.
,
2005; Strumia and Vissani, 2006
)
Y
p
Y
Y
≈
0
.
2
48
+
0
.
0
09
6
×
ln
η
6
.
15
×
10
−
1
0
+
0
.
0
1
3
×
N
4
H
e
ν
N
N
−
3
,
(
9.48
)
w
h
e
r
e
N
4
H
e
ν
N
N
denotes the e
ff
ective number o
f
neutrino s
p
ecies relevant to th
e
4
He abundance. If there were a very light sterile neutrino contributing to th
e
effective number of relativistic degrees of freedom during the BBN, the last
term in Eq.
(
9.48
)
with
N
4
He
ν
NN
=
4 would not be ne
g
li
g
ible. Hence the BBN
theory allows us to constrain not only the baryon-to-photon ratio but also
the number o
f
neutrino species. To determine the value o
f
Y
p
YY
as accurate
ly
a
s
possible, one ma
y
measure the
4
H
e abundance in re
g
ions of hot ionized
g
a
s
f
rom “metal-poor” galaxies where only a relatively small amount of heav
-
i
er elements have been produced throu
g
h the stellar burnin
g
o
f
hydro
g
en
(
Grupe
n
e
tal.
,
2005
)
. So far a lot of data o
n
4
H
e and CNO have been accu-
m
ulated from the most metal-poor
(
HII
)
regions in dwarf galaxies, and the
y
c
on
fi
rm that the small stellar contribution to helium is positivel
y
correlated
with the metal production
(
Nakamura et al.
,
2010
)
. Extrapolating a set o
f
measu
r
e
m
e
n
ts o
f
t
h
e
4
H
e mass fraction to zero metallicity yields the primor
-
d
i
a
l
va
l
ue of
Y
p
Y
Y
as follows
(
Olive and Skillman, 2004
)
:
Y
p
YY
=0
.
249
±
0
.
009
.
Other recent extrapolations to zero metallicit
y
lead us t
o
Y
p
YY
=
0
.
2
4
7
±
0
.
00
1
or
0
.
252
±
0
.
001
(
Izoto
v
et al.
,
2007
)
,o
r
Y
p
YY
=0
.
2
4
8
±
0
.
003
(
Peimbert
et
al.
,
2007
)
. These results are apparently consistent with one another. Fig. 9.
1
s
hows the numerical de
p
endence o
f
Y
p
YY
o
n
η
,
constrained by both the BB
N
a
nd CMB measurements
(
Cyburt
et al
., 2008; Na
k
amur
a
et al.
,
2010
).
A
salient feature of the BBN theor
y
is that it can account not onl
y
for
the
4
He abundance but also for the other light elements
(
such as D,
3
He
9.2 Big Bang Nuc
l
eosynt
h
esis 33
9
η
×
!
Ω
"
J
#
$
$$
−
−
−
−
−
%
&
#
Fi
g
. 9.1 The abundances o
f
D
,
3
H
e
,
4
H
ea
n
d
7
Li e
l
ements pre
d
icte
dby
t
h
eBB
N
theory, where the bands show the ranges at the 95
%
confidence level
(
Cybur
t
e
t
al.
,
2008
)
. The boxes indicate the observational data wit
h
±
2
σ
stat
i
st
i
ca
l
e
rr
o
r
s
(
smaller boxes
)
or wit
h
±
2
σ
s
tatistical and systematic errors
(
larger boxes
)
.The
narrow vertical band stands for the CMB measurement of the cosmic bar
y
on-to-
photon rati
o
η
, while the wider one represents the BBN concordance range. Both o
f
them are given at the 95
%
confidence level
(
Nakamura
et al
., 2010. Wit
hp
ermission
from the Institute of Physics
).
a
n
d
7
L
i
)
whose primordial abundances are rather small but far larger tha
n
the values that would have been i
f
those li
g
ht elements had only been pro
-
d
uced from thermonuclear reactions in stellar interiors. Different from th
e
4
He abundance, which is traditionally
g
iven as the mass
f
raction
Y
p
YY
,
t
h
e
primordial abundances o
f
the other li
g
ht elements are usually presented as
the number fractions. For instance
,
th
e
3
He abundance is described by the
3
He-to-
h
y
d
ro
g
en num
b
er
d
ensity ratio
3
He
/
H
|
p
≡
n
3
H
e
/
n
H
.
Of
particular
i
m
p
ortance is the
p
rimordial deuterium abundance, which can be estimate
d
by the semi-quantitative formula
(
Cirelli
e
ta
l.
,
2005
)
D
H
p
≈
(2
.
75
±
0
.13
)
×
10
−
5
6
.
15
×
10
−
10
η
1
.
6
1
+
0
.
11
N
D
ν
N
N
−
3
(
9.49
)
340 9 Big Bang Nuc
l
eosynt
h
esis an
d
Re
l
ic Neutrinos
with an uncertainty arising mainly from the nuclear cross sections. Assuming
N
D
ν
N
N
=
N
4
He
ν
N
N
≡
N
ν
N
N
,
one may therefore determine or constrain the ma
g
nitude
s
of
η
an
d
N
ν
NN
by combining the measurements of
Y
p
YY
i
nEq.
(
9.48
)
and D
/
H
|
p
i
nEq.
(
9.49
)
. This point is clearly illustrated in Fig. 9.1, where
N
ν
NN
=3h
as
been taken to determine the cosmic bar
y
on-to-photon ratio
η
from the BB
N
d
ata on t
h
eprimor
d
ia
l
Dan
d
4
He abu
n
da
n
ces
.
A
lthou
g
h deuterium can be produced via the reaction
p
+
p
→
D+
e
+
+
ν
e
i
n hydrogen-burning stars, it will quickly be fused into helium. Hence one
believes that there are no astrophysical sources of deuterium
(
Epstein
et al
.
,
1
976
)
. In order to measure the primordial D abundance, one has to detect th
e
h
igh-redshift and low-metallicity gas clouds which are far away and far back in
time and thus have never been a part of stars
(
Grupe
n
et al
.
,
2005
)
. The pres-
ence o
f
deuterium has been measured
f
rom some quasar absorption s
y
stems a
t
h
igh redshifts via its isotope-shifted Lyman
-
α
a
bsorption, and an average o
f
t
h
e measurements
d
one
b
y severa
lg
roups yie
ld
sD
/
H
|
p
=(2
.
82
±
0
.
21
)
×
10
−
5
(
Nakamur
a
et al.
,
2010
)
. We expect that more accurate measurements of th
e
D
abundance might offer the most stringent constraint on the number o
f
n
eutrino species
d
urin
g
t
h
e BBN
.
T
he best determination of the
p
rimordial
7
L
i abundance comes from th
e
m
easurements of hot metal-poor stars in the spheroid
(
Pop II
)
of our galaxy
.
O
ne may extrapolate to zero metallicity to
fi
nd out the ma
g
nitude o
f
the
l
ithium-to-hydro
g
en ratio Li
/
H
|
p
. Current experimental data
y
ield L
i
/
H
|
p
=
(
1
.
7
±
0
.
06
±
0
.
4
4
)
×
10
−
10
, as shown in Fig. 9.1
(
Nakamur
a
et al
.
,
2010
)
.
B
ut o
n
eca
n
see that the stella
rL
i
/
H
m
easu
r
e
m
e
n
ts a
r
e
in
co
n
s
i
ste
n
tw
i
th
the CMB and D
/
H measurements because the
y
apparentl
y
point to different
va
l
ues o
f
η
.
For now t
h
is
l
it
h
ium pro
bl
em is a centra
l
unreso
l
ve
d
issue in t
he
BBN theory, and its solution mi
g
ht involve new physics beyond the
S
Mo
f
particle physics
(
Jedamzik and Pospelov, 2009
).
I
tismuchmoredi
ffi
cult to measure the primordia
l
3
He a
b
un
d
ance, since
the onl
y
available data o
n
3
H
eare
f
rom the solar system and hi
g
h-metallicity
(
HII
)
regions in our galaxy
(
Bani
a
et al., 2002
)
. But one might use th
e
predicted value o
f
the primordia
l
3
He-to-hydrogen ratio
(
i.e.,
3
He
/
H
|
p
as
s
hown in Fig. 9.1
)
to constrain stellar astrophysics.
9.2.3 The Baryon Density and Neutrino
S
pecie
s
T
h
an
k
stot
h
e BBN t
h
eory, it is possi
bl
eto
d
etermine t
h
e cosmic
b
aryon
-
to-p
h
oton ratio
η
a
nd the number o
f
neutrino specie
s
N
ν
NN
f
r
o
m
so
m
e
r
e
li
ab
l
e
m
easurements of the
p
rimordia
l
4
He and D abundances. The value o
f
η
≡
n
B
/n
γ
a
ll
ows us to ca
l
cu
l
ate t
h
e
b
aryon num
b
er
d
ensit
y
n
B
,sincet
h
ep
h
oto
n
n
um
b
er
d
ensit
y
n
γ
≈
4
11
c
m
−
3
i
s already known as given in Eq.
(
9.24
)
with
the CMB tem
p
erature
T
=2
.
7
2
5
K.
W
ehav
e
n
B
≈
0
.
25 m
−
3
today for
η
≈
6
×
10
−
10
.
T
h
e
b
aryons are non-re
l
ativistic to
d
ay, so t
h
eir ener
g
y
d
ensity
i
ssimpl
y
the total mass o
f
nucleons per unit volume
:
9.2 Big Bang Nuc
l
eosynt
h
esis 34
1
ρ
B
=
m
N
n
B
=
m
N
n
γ
η
,
(
9.50
)
whe
r
e
m
N
≈
m
p
≈
m
n
≈
0
.
9
4
G
eV. Then we obtain the ratio o
f
the ener
g
y
d
ensity o
f
baryons to the critical ener
g
y density o
f
the Universe
:
Ω
B
≡
ρ
B
ρ
c
≈
3
.
68
×
10
7
η
h
2
,
(
9.51
)
where
ρ
c
has been given in Eq.
(
9.9
)
. Taking
η
≈
6
×
1
0
−
1
0
an
d
h
≈
7
.
2
to
d
ay at
t
=
t
0
,
for example, we arrive a
t
Ω
B
(
t
0
)
≈
0
.
0
43 toget
h
er wit
h
ρ
B
(
t
0
)
≈
2
.
32
×
10
−
7
G
eV c
m
−
3
≈
4
.
12
×
10
−
2
8
kg
m
−
3
.
T
h
i
s
r
esult
i
s
c
onsistent ver
y
well wit
h
Ω
B
(
t
0
)
=
0
.
043
8
±
0
.
0
013 extracted from the recent
m
easurement of temperature variations in the CMB radiation
(
Nakamura
et al
., 2010
)
. In comparison
,
Ω
r
(
t
0
)
≈
4
.
84
×
1
0
−
5
a
n
d
Ω
m
(
t
0
)
≈
0
.
2
6. W
e
s
ee that
Ω
B
(
t
0
)
Ω
r
(
t
0
)
holds but
Ω
B
(
t
0
)
itself is only about 1
6
.
5% o
f
Ω
m
(
t
0
)
. It is worthwhile to make a remark here: while the relative numbers
o
f
photons, bar
y
ons and antibar
y
ons would have been comparable in the
first nanoseconds after the Big Bang
(
their difference should only be in spi
n
m
ultiplicity factors
)
, most of the nucleons and antinucleons must have later
d
isappeared due to their mutual annihilation, leavin
g
a tiny — about one par
t
per billion — excess of nucleons as the matter of the everyday world
(
Perkins,
2
009
)
. In other words, there is no primordial antimatter surviving today in
the observable Universe. This cosmolo
g
ical matter-antimatter asymmetry will
be discussed in detail in Chapter 11
.
O
nce the cosmic bar
y
on-to-photon ratio
η
i
s
k
nown, t
h
eprimor
d
ia
ld
eu
-
terium or helium abundance can be fixed to a quite narrow range by the BB
N
theory
(
see Fig. 9.1 for illustration
)
. As shown in Eqs.
(
9.48
)
and
(
9.49
)
,
a
s
imple comparison between the predicted and measured values o
f
Y
p
YY
o
rD
/
H
|
p
a
llows us to determine or constrain the effective number of neutrino s
p
ecies
.
N
ote t
h
at t
h
e
h
e
l
ium an
dd
euterium curves in Fi
g
.9.1arep
l
otte
db
yta
k
in
g
N
eff
ν
N
N
= 3 in the limit of an instantaneous neutrino decoupling
(
Cybur
t
et
a
l
.
,
2008
)
. If small corrections for non-equilibrium neutrino heating are take
n
i
n
to accou
n
t
in
the the
rm
al evolut
i
o
n
below
T
∼
1
MeV, one
h
a
s
N
eff
ν
NN
≈
3
.
04
(
Seljak and Zaldarriaga, 1996; Mangano
et al
., 2002
)
. Note also that
N
eff
ν
NN
m
atters in the BBN era
(
fo
r
T<
m
e
)
via the effective number of degrees of
f
r
eedo
m
g
∗
≈
2
1
.
6
81
+
0
.
227
N
eff
ν
NN
−
3
,
(
9.52
)
w
h
ic
h
is equiva
l
ent t
o
g
∗
g
iven in Eq.
(
9.41
)
. For instance
,
g
∗
≈
3
.
8
2wou
ld
ho
l
d
i
f
N
e
ff
ν
NN
=4wereta
k
en, as compare
d
wit
h
g
∗
≈
3
.
3
6
f
or
N
eff
ν
NN
=3.Henc
e
the presence of extra neutrino flavors
(
or of any other relativistic species
)
a
t
th
i
st
im
e would e
n
ha
n
ce
g
∗
,
leadin
g
toalar
g
er
f
reeze-out temperature
T
fr
T
T
,
a
lar
g
er neutron-to-proton rati
o
n
n
/
n
p
and a lar
g
er
4
H
e
m
ass f
r
act
i
o
n
Y
p
YY
as
one can easily see from Eqs.
(
9.32
)
,
(
9.43
)
and
(
9.47
)
. This observation was
ori
g
inally used to constrain the number o
f
li
g
ht neutrino
f
amilies be
f
ore the
L
EP ex
p
eriment demonstrated
N
ν
N
N
=3toahi
g
hde
g
ree o
f
accuracy
f
or the
SM of electroweak interactions
(
Nakamura
e
tal
.
,
2010
).
342 9 Big Bang Nuc
l
eosynt
h
esis an
d
Re
l
ic Neutrinos
Th
ea
b
ove
d
iscussions i
ll
ustrate t
h
at t
h
e BBN provi
d
es us wit
h
an inter
-
estin
g
play
g
round to enjoy the interplay between particle physics and cos-
m
o
l
ogy. Just as one may use t
h
eo
b
serve
dh
e
l
ium an
dd
euterium a
b
un
d
ances
to probe the e
ff
ective number o
f
de
g
rees o
ff
reedo
m
g
∗
,anyc
h
an
g
es in t
he
s
tron
g
, weak, electroma
g
netic or
g
ravitational couplin
g
constants can be sim
-
i
larly constrained
(
Nakamura
et al
., 2010
)
. Furthermore, the constraints on
the e
ff
ective number o
f
neutrino species can be translated into the limits o
n
other t
y
pes of particles or particle masses that would affect the expansio
n
rate of the Universe during the BBN. For the sake of simplicity, however,
we do not
g
ointodetailso
f
concrete new physics models and their possibl
e
c
onse
q
uences on the BBN in this book.
9
.3 Poss
i
ble Wa
y
s to Detect Rel
i
cNeutr
i
nos
The Bi
g
Ban
g
model o
f
cosmolo
g
y predicts the existence o
f
a
C
ν
Bcompose
d
of relic neutrinos which were decou
p
led from matter and radiation at about
T
fr
TT
1M
e
V
or
t
0
.
7
4 s after the Big Bang. It is quite analogous to the
well-known
C
MB, whose
f
ormation was at a time o
f
about
3
.
8
×
10
5
y
ears
a
fter the Big Bang
(
i.e., when the photons were decoupled from matter a
t
a
temperature of about 0.26 eV
)
. At present the observational evidence for
this
C
ν
B is rather indirect, comin
g
mainly
f
rom the measurements o
f
the
primordial abundances of light elements, the anisotropies in the CMB and
the lar
g
e-scale structures o
f
the cosmos. Is it possible to directly detect the
C
ν
B
by means o
f
current experimental technolo
g
ies
?
9.3.1
C
osmic Neutrino Background
In
S
ection 9.1.5 we have shown that the lar
g
e-scale properties o
f
the
C
ν
B
a
re closel
y
related to those of the CMB. In particular, the temperatures o
r
n
umber densities o
f
relic neutrinos and photons are related to each other
:
T
ν
TT
=
4
11
1
/
3
T
γ
TT
,n
ν
=
9
11
n
γ
.
(
9.53
)
Give
n
T
γ
TT
≈
2
.
7
25 K and
n
γ
≈
4
11 cm
−
3
for the CMB radiation toda
y
,th
e
c
orresponding temperature and number density of the
C
ν
Btu
rn
out to be
T
ν
TT
≈
1
.
9
45 K an
d
n
ν
≈
336
c
m
−
3
.A
s a consequence, the avera
g
e three-
m
omentum of each relic neutrino is very small
(
Ringward, 2009
)
:
p
ν
=
3
T
ν
TT
≈
5
.
8K
≈
5
×
1
0
−
4
eV
,
(
9.54
)
i
mplyin
g
that at least two mass ei
g
enstates o
f
the relic neutrinos are alread
y
n
on-relativistic toda
y
no matter whether the neutrino mass spectrum has
a
normal hierarchy or an inverted hierarchy
(
see Section 8.4.1 for a similar
9.3 Possi
bl
e Ways to Detect Re
l
ic Neutrinos 34
3
d
iscussion
)
.The
C
ν
B
is naive
l
y expecte
d
to
b
e
h
omo
g
eneous an
d
isotropic on
l
ar
g
e scales, but it is somewhat subject to the
g
ravitational clusterin
g
effec
t
d
ue to the existence of cold dark matter and baryonic structures. Hence som
e
l
ocal overdensities mi
g
ht appear in the
C
ν
B
as compare
d
wit
h
t
h
estan
d
ar
d
value given above, and the momentum distribution might more or less deviat
e
f
rom the one following from the Fermi-Dirac distribution
.
T
he presence o
f
the
C
ν
B
can a
ff
ect the evolution o
fC
MB anisotropies an
d
the growth of matter perturbations. For instance, relativistic relic neutrino
s
c
ontributed to the energy density of radiation and thus had an impact on
thetimeo
f
matter-radiation equality. The
f
ree-streamin
g
massive neutrinos
c
ould also suppress the growth of structures on small scales. Similar to CM
B
a
nisotropies,
C
ν
B
anisotropies
h
ave
b
een ca
l
cu
l
ate
d
a
l
t
h
ou
gh
it is extreme
l
y
d
i
ffi
cult to detect them even in the
f
ar
f
uture. I
f
neutrinos were massless
,
the
p
ower s
p
ectrum of Cν
B
fluctuations would closely resemble the usual
CMB spectrum without the baryon-photon acoustic oscillations
(
Hu
et al
.
,
1
995
)
. The calculation of the primary C
ν
B
s
p
ectrum is rather com
p
licated
f
or massive neutrinos
,
and the
C
ν
B
anisotropy can
b
e quite sensitive to t
he
a
bsolute values of neutrino masses
(
Hannestad and Brandbyge, 2009
).
9.3.2 Direct Detection o
f
Relic Neutrinos
A
sindicatedinEq.
(
9.54
)
, relic neutrinos have little kinetic energies today.
It is extremely difficult to detect such low-energy neutrinos because the cros
s
s
ections o
f
their interactions with matter within a detector are terribl
y
sup-
pressed. Nevertheless, it is ver
y
important to directl
y
verif
y
the existenc
e
of the
C
ν
B
in order to test the neutrino aspects of the Big Bang model o
f
c
osmolo
g
y. Here we
g
ive a brie
f
overview o
f
a
f
ew proposals
f
or the direc
t
d
etection of the
C
ν
B
in the present epoch and in our local neighborhood.
(
1
)
R
e
l
ic neutrino capture on
β
-d
ecaying nuc
l
ei
.
T
h
e most promisin
g
m
ethod
f
or detectin
g
the
C
ν
B
is to search
f
or a peak, which is related t
o
the capture of a relic electron neutrino, in the energy spectrum of a beta
d
ecay. This idea was
fi
rst proposed by
S
teven Weinber
g
alon
g
time a
go
(
Weinberg, 1962; Irvine and Humphreys, 1983
)
, and has recently attracted
s
ome interest
(
Cocco
e
tal.
,
2007
,
2009
;
Lazauskas et al
.
,
2008
;
Blennow
,
2008
;
Kaboth
et al
., 2010;
L
i
et al
., 2010
)
. The point can be made clear as follows
.
Let us co
n
s
i
de
r
a
n
uc
l
eus
N
which can naturally under
g
o the beta deca
y
N
→
N
+
e
−
+
ν
e
with an energy releas
e
Q
β
=
m
N
−
m
N
−
m
e
>
0
in th
e
l
imit of vanishing neutrino masses
(
i.e.
,
m
i
→
0f
o
r
i
=
1
,
2
,
3
)
. Then we see
that the neutrino ca
p
ture reactio
n
ν
e
+
N
→
N
+
e
−
(
9.55
)
h
as a salient
f
eature: it can take place with no threshold on the incident neu-
tr
i
no ener
gy
E
ν
i
(
for each mass eigenstat
e
ν
i
in
ν
e
)
. In contrast, the reaction
ν
e
+
N
→
N
+
e
+
is subject to a threshold on the incoming antineutrino
344 9 Big Bang Nuc
l
eosynt
h
esis an
d
Re
l
ic Neutrinos
ener
g
y
E
ν
i
.
The number of events for the neutrino capture in Eq.
(
9.55
)
i
s
determined by the product of the cross section of the reaction itself and the
i
flux of relic ν
e
n
eutrinos, w
h
ic
hd
epen
d
respective
l
yon
1
/v
ν
i
a
n
d
v
ν
i
wit
h
v
ν
i
∼
3
T
ν
T
T
/m
i
b
eing the modulus of the velocity o
f
ν
i
(
Lazauska
s
et al.
, 2008
).
H
ence it conver
g
es to a
fi
nite value even
f
o
r
v
ν
i
→
0
o
r
E
ν
i
→
0an
d
ca
n
be reasonably large if a sufficient amount of th
e
β
-decaying target materia
l
i
s prepared
(
Cocc
o
et al.
, 2007; Lazaus
k
as
et al
.
,
2008
)
. In the extreme case
w
i
th
m
i
→
0and
E
ν
i
→
0
,the
ν
e
n
eut
rin
oco
n
t
ri
butes to t
hi
s
r
eact
i
o
n
v
i
a
i
ts correct lepton number and the final-state electron exactly possesses the
β
-d
ecay en
d
point ener
g
y
Q
β
.G
iven
fi
nite values o
f
neutrino masse
s
m
i
,
h
ow
-
ever, the monoener
g
etic electron’s kinetic ener
g
yi
s
Q
β
+
E
ν
i
Q
β
+
m
i
f
or each neutrino mass eigenstat
e
ν
i
. In comparison, the non-monoenergetic
electron emitted
f
rom the beta deca
y
N
→
N
+
e
−
+
ν
e
ca
rri
es a k
in
et
i
c
ener
g
y
Q
β
−
E
ν
i
Q
β
−
m
i
f
or each
ν
i
i
f the recoil of the final-state nucleu
s
N
is ne
g
lected. So there is a
g
ap equal to or lar
g
er than 2m
i
between th
e
e
l
ectron
’
s
k
inetic energy in ν
e
+
N
→
N
+
e
−
an
d
t
h
at in
N
→
N
+
e
−
+
ν
e
(
Cocco
et al
.
,
2007
)
7
.
T
h
is o
b
servation imp
l
ies t
h
at it is possi
bl
e, at
l
east in
principle, to distinguish the neutrino capture reaction from its correspond-
i
ng beta decay by measuring the energy spectrum of final-state electrons t
o
a
n unprecedentedly hi
g
hde
g
ree o
f
accuracy characterized by the neutrino
m
asses. One ma
y
therefore has a chance to directl
y
detect the presence o
f
the
C
ν
Bint
h
is way
.
T
he ca
p
ture rate o
f
relic electron neutrinos via
ν
e
+
N
→
N
+
e
−
ca
n
be
given as
(
Lazauska
s
e
tal
.
,
2008
;
Blennow
,
2008
;
L
i
e
tal
.
2
010
)
N
(
i
)
C
NN
ν
B
≈
6
.
5
|
V
ei
VV
|
2
n
ν
i
¯
n
ν
i
yr
−
1
M
Ci
−
1
,
(
9.56
)
w
h
ere
V
ei
V
V
d
enotes t
h
e neutrino mixing matrix e
l
ement an
d
|
V
ei
VV
|
2
m
eas
ur
es
t
h
ee
l
ect
r
o
nn
eut
rin
oco
n
te
n
tof
ν
i
(
for
i
=1
,
2
,
3),
n
ν
i
/
¯
n
ν
i
is t
h
e
r
at
i
oof
the relic neutrino densit
y
on the Earth to the mean relic neutrino densit
y
in
the Universe for each neutrino mass eigenstate. With the help of Eq.
(
9.53
)
,
we have ¯
n
ν
i
=
n
ν
/
6
≈
56
c
m
−
3
.
We expect t
h
a
t
n
ν
i
/
¯
n
ν
i
1
holds
in
v
i
ew
of the gravitational clustering of relic neutrinos
(
Ringwald and Wong, 2004
)
.
Taking
N
t
o be the tritium
(
i.e., ν
e
+
3
H
→
3
He
+
e
−
)
, for example, one
has
Q
β
=18
.
6 keV and 1 M
C
isourceo
f
tritium is about 100
g
.Inthi
s
c
ase the capture rate should be reasonably lar
g
e
f
or a lon
g
runnin
g
time
of the ex
p
eriment. The main
p
roblem is whether the Cν
B
signal would be
d
etectable
g
iven the overwhelmin
g
rate o
f
the beta decay.
Of
course, th
e
s
i
g
nal-to-noise ratio critically depends o
n
m
i
,
|
V
ei
VV
|
and the ener
g
y resolution
7
If
the mass o
f
the li
g
htest neutrino is belo
w
p
ν
g
iven in Eq.
(
9.54
)(
i.e.
,
it remains hot or relativistic today
)
,thenitsenergycanbeexpresseda
s
E
ν
i
≈
p
ν
2
+
m
2
i
,
which is o
f
O
(
p
ν
)
and hence has little impact on the overall electron
energy spectrum under discussion (Li
e
ta
l
.
,
2010).
9.3 Possi
bl
e Ways to Detect Re
l
ic Neutrinos 34
5
o
f
the detection apparatus
Δ
(
Cocc
o
et al
.
,
2007; Lazaus
k
a
s
et al
.
, 2008
;
Blennow
,
2008
;
Kabot
h
e
tal.
,
2010
;
Li
e
tal., 2010
)
. But this ratio relie
s
n
either on the value of
Q
β
n
or on the nuclear matrix elements for the final-
s
tate e
l
ectrons to
l
ie wit
h
in t
h
eener
g
y reso
l
ution interva
l
Δ
j
ust
b
e
l
ow t
he
end
p
oint. Give
n
m
i
<
Δ
,
the corresponding signal-to-noise ratio is expecte
d
to be
N
(
i
)
C
NN
ν
B
N
(
i
)
β
NN
≈
6
π
2
n
ν
i
Δ
3
≈
2
.
5
×
10
−
1
1
n
ν
i
¯
n
ν
i
1
e
V
Δ
3
.
(
9.57
)
This number is hopelessly small, unless it could be si
g
ni
fi
cantly enhanced by
a
n extreme gravitational clustering effect
.
I
t is unrea
l
istic to app
l
yt
h
ea
b
ove met
h
o
d
to a
ll
t
h
e present
l
y-
d
eve
l
ope
d
β
-
decay experiments, including the KATRIN experiment
(
Otten and Wein-
h
eimer, 2008
)
, for two obvious reasons:
(
a
)
the amount of their target materia
l
i
s very small, leading to a tiny total capture rate of relic neutrinos; and
(
b
)
their energy resolution is very poor
(
Δ
m
i
)
, such that the signal-to-noise
ratio is too small to claim a discovery
(
Ringwald, 2009
).
If
the
C
ν
B
consists of both active neutrinos
(
ν
1
,
ν
2
,
ν
3
)
and sterile neutri
-
n
os
(
ν
4
ν
,
ν
5
,
···
)
, the latter may also leave an imprint on the electron energy
s
pectrum in the capture of relic electron neutrinos by means of radioactiv
e
beta-decayin
g
nuclei. To be more explicit, let us assume that the masses o
f
s
terile neutrinos are of
O
(0
.
1)
eV and somewhat larger than the absolut
e
m
ass scale of three active neutrinos. In this case we have a few very genera
l
expectations about the direct laborator
y
detection o
f
the
C
ν
Ba
n
d
i
ts ste
r
-
i
le component
(
Li et al.
,
2010
)
. First, the signal of the sterile component of
the
C
ν
B is on the right-hand side of the electron energy spectrum as com-
pared with the si
g
nal o
f
the active component o
f
the
C
ν
B
.Theirse
p
aration
i
s measured by their mass differences. Second, the rate of events for the sig
-
n
al o
f
relic sterile neutrinos is crucially dependent on the ma
g
nitude o
f
their
m
ixin
g
with active neutrinos. Hence a lar
g
er value o
f
|
V
ei
VV
|
(
fo
r
i
4)
leads
to a higher rate of signal events. Third, whether a signal can be separate
d
f
rom its back
g
round depends on the
fi
nite ener
g
y resolutio
n
Δ
i
n
a
r
eal
i
st
i
c
experiment. In
g
eneral
,
Δ
m
i
/
2(
for
i
=
1
,
2
,
···
)
is required to detect th
e
C
ν
B
via a neutrino capture reaction
.
(
2
)
A
Cavendish-t
y
pe torsion balance
f
or relic neutrinos
.W
earenotatres
t
with res
p
ect to the almost isotro
p
ic CMB, nor to the almost isotro
p
ic C
ν
B
because the Earth is moving through both of them. The coherent scattering o
f
a
relic neutrino
fl
ux o
ff
the tar
g
et matter in a terrestrial detector may
g
ive ris
e
to a mechanical force
(
Shvartsman et al
.
, 1982; Smith and Lewin, 1983
)
,which
i
s possibly detectable in a Cavendish-type torsion balance by searching for
a
n annual modulation of the signal
(
Hagmann, 1999
)
. The point is that relic
n
eutrinos have a macroscopic de Broglie wavelength
λ
ν
=
1
/T
ν
T
T
≈
0
.12 cm
,
whe
r
e
T
ν
TT
≈
1
.
945 K has been input. So a terrestrial detector with a mas
s
d
ensit
y
ρ
a
n
da
lin
ea
r
s
iz
e
r
<
λ
ν
ma
y
experience a tin
y
acceleration induced
by the neutrino wind due to the coherent neutrino-nucleon scattering effect.