Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
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316 8 Ultrahigh-energy
C
osmic Neutrinos
by means of the known gamma ray spectrum
(
Vissani, 2006; Vissani an
d
Villante, 2008; Villante and Vissani, 2008
)
.
Let us assume that equal numbers o
f
π
+
,
π
−
an
d
π
0
e
vents
h
ave
b
een
produced
f
rom a
S
NR. Then VHE
g
amma rays can ori
g
inate
f
rom the deca
y
m
ode
π
0
→
γ
+
γ
,
and their flux
φ
γ
(
E
)
is related to the flux of neutral pions
φ
π
0
(
E
)
in the following way
(
Vissani, 2006
)
8
:
φ
γ
(
E
)
=2
∞
E
d
E
φ
π
0
(
E
)
E
;
(
8.30
)
or equiva
l
ent
l
y,
d
φ
γ
(
E
)
d
E
=
−
2
φ
π
0
(
E
)
/
E, implyin
g
that the
g
amma ray
fl
ux
h
as to be strictly decreasing. On the other hand, VHE neutrinos can originate
f
rom the deca
y
mode
π
+
→
μ
+
+
ν
μ
. Similar to Eq.
(
8.30
)
,th
e
ν
μ
a
n
d
π
+
fluxes are related to each other thou
g
h
φ
ν
μ
(
E
)=
κ
∞
κE
d
E
φ
π
+
(
E
)
E
,
(
8.31
)
where κ
≡
1
/
(1
−
r
)
wit
h
r
≡
m
2
μ
/
m
2
π
≈
0
.
5
73. In the extremel
y
relativistic
l
imi
t
E
ν
μ
a
n
d
E
μ
+
l
ie in t
h
e ranges
0
E
ν
μ
E
π
+
(
1
−
r
)
an
d
r
E
π
+
E
μ
+
E
π
+
,
respective
l
y. Note t
h
at t
h
e approximate isospin-invariant
d
istri
b
utio
n
o
fp
ions im
p
lie
s
φ
π
0
≈
φ
π
+
≈
φ
π
−
.
Then a comparison between Eqs.
(
8.30
)
a
nd
(
8.31
)
leads us to the relationship
φ
ν
μ
(
E
)=
κ
φ
γ
(
κE
)
/
2
. In addition
,
φ
μ
+
c
an be read off from Eq.
(
8.31
)
by replacing its integral’s interval with
E
μ
+
∈
[
rE
π
+
,E
π
+
]
;namely,
φ
μ
+
(
E
)=
κ
[
φ
γ
(
E
)
−
φ
γ
(
E
/
r
)]
/
2
.T
h
e
μ
+
d
eca
y
pro
d
uces
ν
μ
a
n
d
ν
e
, whose flux is given by
(
Vissani, 2006
)
φ
ν
(
E
ν
)
=
1
0
d
y
y
φ
μ
+
(
E
μ
+
)
g
0
(
y
)
−
g
1
(
y
)
P
μ
(
E
μ
+
)
,
(
8.32
)
w
h
ere t
h
esu
b
script
ν
de
n
otes
ν
e
or
ν
μ
,
y
≡
E
ν
/E
μ
+
,
g
0
(
y
)
=2
(1
−
3
y
2
+2
y
3
)
a
n
d
g
1
(
y
)
=
−
2
(
1
−
6
y
+9
y
2
−
4
y
3
)
fo
r
ν
=
ν
e
,o
r
g
0
(
y
)
=
(5
−
9
y
2
+4
y
3
)
/
3an
d
g
1
(
y
)
=
(1
−
9
y
2
+8
y
3
)
/
3for
ν
=
ν
μ
,
and
P
μ
+
describes the
μ
+
p
olarizatio
n
a
vera
g
e
d
over t
he
π
+
d
i
st
ri
but
i
o
n:
P
μ
+
(
E
)
×
φ
μ
+
(
E
)
=
−
κ
2
φ
γ
(
E
)+
φ
γ
(
E
/r
)
+
κ
2
r
E/r
E
d
E
φ
γ
(
E
)
E
.
(
8.33
)
As
f
or
π
−
→
μ
−
+
ν
μ
a
n
d
μ
−
→
e
−
+
ν
μ
+
ν
e
d
ecays, t
h
e correspon
d
ing
ν
μ
,
ν
μ
a
n
d
ν
e
fl
uxes can similarly be given by Eqs.
(
8.31
)
and
(
8. 32
)
.Thuson
e
8
W
e have assumed that the VHE gamma ray flux is not significantly absorbed by
t
h
e source. Hence t
h
emet
h
o
dd
escri
b
e
dh
ere is not
d
irect
ly
app
l
ica
bl
etoanum
b
er
of possible galactic sources of VHE photons and neutrinos such as microquasars
t
h
at are intrinsica
ll
y non-transparent, or to possi
bl
e extraga
l
actic sources since t
h
e
in
f
rared photon back
g
round absorbs VHE
g
amma rays i
f
their ener
g
ies are above
about 10 TeV (Aharonia
n
et a
l.
, 2006b; Vissani, 2006)
.
8
.4 Neutrinos and Multi-messenger Astronomy 317
m
a
y
obtain the
fl
uxes o
f
VHE neutrinos and antineutrinos in terms o
f
that
of VHE
g
amma rays from a
g
alactic SNR. Two comments are in order.
•
O
ne may include the contribution o
f
th
e
η
→
γ
+
γ
d
ecay to t
h
e VHE
p
hoton
fl
ux and those o
f
le
p
toni
c
K
±
deca
y
s to the VHE neutrino
fl
uxes
in a more careful analysis
(
Vissani, 2006; Villante and Vissani, 2008
)
.
•
O
ne may incorporate neutrino oscillations into the above
f
ormulas wit
h
t
he re
p
lacement
φ
ν
μ
→
φ
ν
μ
P
(
ν
μ
→
ν
μ
)
+
φ
ν
e
P
(
ν
e
→
ν
μ
)
,where
P
(
ν
α
→
ν
β
ν
)
(
for α,
β
=
e
,
μ
,
τ
)
have been given in Eq.
(
8.7
)
.Thesam
e
t
reatment is valid for antineutrinos
(
Vissani and Villante, 2008
).
It is then possible to compute the fluxes of VHE neutrinos and antineutrino
s
with the help of the observed VHE gamma rays
(
Vissani, 2006; Villante an
d
Vissani
,
2008
;
Vissani and Villante
,
2008
;
Morlin
o
e
tal
.
,
2009
)
.
T
he astrophysical sources of extragalactic cosmic rays include AGN an
d
G
RBs. For example, the
fi
reball producin
g
a
G
RB converts a
f
raction o
f
a
s
olar mass into the acceleration of electrons which is seen as s
y
nchrotro
n
photons
(
Halzen, 2009
)
. Shocks in the expanding GRB fireball may convert
rou
g
hly the same amount o
f
ener
g
y into the acceleration o
f
protons that ar
e
observed as UHE cosmic rays
(
Waxman, 1995
)
. In this scenario cosmic ray
s
a
nd synchrotron photons coexist be
f
ore the
fi
reball reaches transparency and
produces the observed
G
RB displa
y
. Their interactions via the
Δ
+
-
r
eso
n
a
n
ce
p
r
ocesses
p
+
γ
→
Δ
+
→
π
0
+
p
an
d
p
+
γ
→
Δ
+
→
π
+
+
n
can give rise t
o
n
eutra
l
an
d
c
h
ar
g
e
d
pions, respective
l
y, wit
h
pro
b
a
b
i
l
ities 2
/
3
an
d1
/
3
.
U
HE
g
amma rays and neutrinos are then produced fro
m
π
0
→
γ
+
γ
,
π
+
→
μ
+
+
ν
μ
a
n
d
μ
+
→
e
+
+
ν
μ
+
ν
e
decays. As discussed in Section 8.1.2, about 1
/
4
(
o
r
1/
2
)
of the pion’s energy is taken away by each neutrino
(
or photon
)
.Thenwe
h
av
e
x
ν
≡
E
ν
/E
p
E
≈
0
.
2
5
E
π
/
E
p
E
a
n
d
x
γ
≡
E
γ
E
/E
p
E
≈
0
.
5
E
π
/
E
p
E
,
where
E
π
/
E
p
E
≈
π
/
(
1
+
π
)
≈
0
.22 wit
h
π
≡
E
π
/
E
n
≈
0
.
28
measures t
h
e
a
vera
g
eener
g
ytrans
f
erred
f
rom the proton to the pion. The UHE neutrino
a
nd photon fluxes produced by a GRB are therefore given b
y
d
φ
ν
d
E
=1
×
1
3
×
1
x
ν
d
φ
p
d
E
p
E
E
p
=
E/x
ν
,
d
φ
γ
d
E
=2
×
2
3
×
1
x
γ
d
φ
p
d
E
p
E
E
p
=
E
/x
γ
,
(
8.34
)
where
φ
ν
(
=
φ
e
=
φ
μ
=
φ
τ
)
stands for the sum of neutrino and antineutrin
o
fl
uxes o
f
each
fl
avor which will not be distin
g
uished at a neutrino telescope
.
P
rovided the energy spectrum of primary protons is taken to be
d
φ
p
d
E
p
E
∝
E
−
2
p
E
,
then Eq.
(
8.34
)
leads us to the relatio
n
d
φ
γ
d
E
=8
d
φ
ν
d
E
.
It is we
ll k
nown t
h
a
t
n
eutrino oscillations over cosmic distances approximatel
yy
ield equal fluxes
f
or three flavors. Given some details of the source and the detector, one shoul
d
318 8 Ultrahigh-energy
C
osmic Neutrinos
be able to evaluate the
fl
ux o
fU
HE neutrinos and the number o
f
events to be
d
etected
(
Halzen, 2009
)
. It might be impossible to observe the corresponding
U
HE gamma rays because they would be absorbed by the CMB on their wa
y
f
rom an extra
g
alactic source to the Earth
.
8
.4.
3
Neutrinos and
G
ravitational Waves
A
s a classical field theory of gravitation, Einstein’s general relativity predict
s
the existence o
fg
ravitational waves emitted
f
rom massive objects under
g
oin
g
l
ar
g
e accelerations. Typical examples include the binary systems o
f
compac
t
o
b
jects
l
i
k
e neutron stars an
dbl
ac
kh
o
l
es, or u
l
tra-compact
b
inaries suc
h
a
s double white dwarfs and ultra-compact X-ray binaries
(
Avignon
e
et al.
,
2
008
)
. Many attempts have been made to verify the existence of gravitationa
l
waves. In particu
l
ar, Russe
ll
Hu
l
se an
d
Josep
h
Tay
l
or receive
d
t
h
e 1993 No
b
e
l
P
rize
f
or their indirect detection o
fg
ravitational waves throu
g
h the ener
g
y
l
oss of the binary pulsar PSR 1913+16
(
Hulse and Taylor, 1975
)
. The direct
observation of gravitational waves is still a big challenge in astrophysics,
a
lthou
g
h the experimental techniques are becomin
g
more mature than be
f
ore
.
S
ince supernova explosions are believed to be the sources of long-timescale
GRBs, the observation of gravitational waves expected from such violen
t
events may a
ll
ow us to
b
etter un
d
erstan
d
t
h
e
d
ynamica
l
processes
g
ivin
g
ris
e
to GRBs. In the forthcoming era of multi-messenger astronomy it will b
e
particu
l
ar
l
y interestin
g
to
d
etect
h
i
gh
-ener
g
y neutrinos in coinci
d
ence wit
h
g
ravitational waves produced
f
rom supernovae and
G
RBs. Both messen
g
ers
c
an escape very dense media and travel over cosmological distances withou
t
bein
g
absorbed, pointin
g
back to their common sources and carryin
g
in
f
orma-
tion from the innermost regions of those astrophysical engines
(
Van Elewyck,
2
009
)
. They could also reveal possible hidden sources that are unable to b
e
s
een by means o
f
hi
g
h-ener
g
y
g
amma rays. Indeed, neutrino bursts associate
d
withthesamesourceas
g
ravitational waves can lead to some effects in the
gravitational wave antennas, but the contrary is not true
(
Castagnoli
e
ta
l
.
,
1
978
)
. The point is simply that the velocity of a massive neutrino is slightly
s
maller than that of the massless
g
raviton and hence results in a time dela
y
between their signals to be detected on the Earth. It is therefore meaningfu
l
a
nd use
f
ul to establish a correlation in time amon
g
neutrino and
g
ravitational
wave detectors for a given astrophysical source. On the other hand, the de
-
tection of gravitational waves and high-energy neutrinos originating from the
earl
y
Universe will provide a unique source o
f
in
f
ormation about
f
undamental
ph
y
sics at work in the earl
y
Universe
.
T
he
f
easibility o
f
searchin
gf
or hi
g
h-ener
g
y neutrinos in coincidence wit
h
g
ravitational waves is under discussion. In principle, a neutrino telescope ca
n
a
ccurately determine the time and direction of high-energy neutrino events,
while a network o
fg
ravitational wave detectors can also provide timin
g
and
d
irectional information for gravitational wave bursts
(
Van Elewyck, 2009
)
.
Combining the measurements from these totally independent detectors will
R
e
f
erences 31
9
all
ow one to ac
h
ieve ro
b
ust
b
ac
kg
roun
d
rejection an
d
muc
hb
etter sensi
-
tivities.
A
t
y
pical example is to combine the
A
NT
A
RES neutrino telescop
e
(
Carr, 2008
)
with the VIRGO and LIGO interferometers for gravitationa
l
waves
(
Acernese
et al
.
,
2008;
A
bbot
t
et al
.
,
2008
)
. They share the challeng
e
to search for faint and rare signals on top of abundant noise or backgroun
d
events, and the observation of coincident triggers would provide strong ev-
i
dence
f
or a
g
ravitational wave burst and a neutrino burst comin
gf
rom
a
c
ommon source
(
Van Elewyck, 2009; Pradier, 2009
)
.
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eference
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hou, S., 2008, Ph
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9
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9
B
i
gBangNucleosynthes
i
s and Rel
i
cNeutr
i
nos
The concerns of cosmolo
g
y include the birth of the Universe, its evolution, its
s
tructure, and its ultimate fate. At present the most convincing theoretica
l
para
d
i
g
mincosmo
l
o
g
yist
h
eso-ca
ll
e
d
Bi
g
Ban
g
mo
d
e
l
,w
h
ic
h
attri
b
utes t
he
ori
g
in of the Universe to an extremely hot and dense plasma that was hi
g
hl
y
i
sotropic and homogeneous. In this intriguing picture the evolution of the
earl
y
Universe underwent a number o
f
crucial events, such as the in
fl
ation,
q
uark-baryon transition, neutrino decoupling, nucleosynthesis and photon de
-
c
oupling. The Big Bang nucleosynthesis
(
BBN
)
has so far provided us wit
h
themostreliableprobeo
f
the earl
y
Universe based on the well-establishe
d
knowledge of particle physics and nuclear physics. In this chapter we shal
l
m
ake use of the standard Big Bang model to describe the irreplaceable role of
n
eutrinos in the histor
y
o
f
the Universe. In particular, their contributions t
o
the energy density of the Universe, their effects on the BBN and their prop
-
erties as the cosmic neutrino background
(C
ν
B)
will be discussed in some
d
etail. Possible ways and experimental challen
g
es to directly detect these Bi
g
Bang relic neutrinos will also be introduced
.
9
.1 Neutrinos in the Early Univers
e
The Bi
g
Ban
g
model o
f
cosmolo
g
y, which was
fi
rst
f
ormulated by
G
eor
ge
Gamow and his collaborators in the 1940’s
(
Gamow, 1946; Alpher
e
tal
.
,
1
948
)
, is based on Einstein’s general relativity and the cosmological principle
.
T
h
e
l
atter states t
h
at t
h
eUniversewas
h
omo
g
eneous an
d
isotropic at ear
ly
times and on large scales, implying that all spatial positions in the Universe
s
hould be essentially equivalent. Instead o
fg
oin
g
into details o
f
the standard
Bi
g
Ban
g
model o
f
cosmolo
g
y, which has been described in an excellent wa
y
i
nsometextbooks
(
see, e.g., Kolb and Turner, 1990; Weinberg, 2008
)
,w
e
s
hall brie
fl
y introduce some o
f
its key in
g
redients and consequences relevan
t
to the expansion o
f
the earl
y
Universe and the interpla
y
o
f
matter, radiation
a
nd vacuum energy up to the time of neutrino decoupling.
Z.-Z. Xing et al., Neutrinos in Particle Physics, Astronomy and Cosmology
© Zhejiang University Press, Hangzhou and Springer-Verlag Berlin Heidelberg 2011
324 9 Big Bang Nuc
l
eosynt
h
esis an
d
Re
l
ic Neutrinos
9.1.1 Hubble’s Law and the Fr
i
edmann Equat
i
ons
One of the seminal discoveries in the histor
y
of science is Edwin Hubble’s
observation that galaxies have been receding from each other at a velocity
v
w
h
ic
h
is
l
inear
ly
proportiona
l
to t
h
eir
d
istance
D
(
Hubble, 1929
)
:
v
=
HD
wit
h
H
bein
g
the time-dependent Hubble parameter. The present-day value
of
H
,k
nown as t
h
eHu
bbl
e constant
H
0
HH
≡
H
(
t
0
)
at
t
=
t
0
,h
as
b
een measure
d
to a
g
ood de
g
ree o
f
accuracy:
H
0
HH
=
100
h
k
m
s
−
1
Mp
c
−
1
w
i
th
h
=
0
.
72
±
0
.
03
(
Nakamur
a
e
tal., 2010
)
1
. The ph
y
sical coordinate distance D from the Earth
to a
d
istant ga
l
axy at tim
e
t
ca
n
be w
ri
tte
n
as
D
(
t
)
=
rR
(
t
)
,wher
e
r
de
n
otes
the time-independent comovin
g
coordinate distance measured in a re
f
erenc
e
f
rame moving together with the expansion of the Universe, an
d
R
(
t
)
is th
e
c
osmological scale parameter which determines proper distances in terms o
f
the comoving coordinates. Given the cosmological principle
(
i.e., homogeneit
y
a
nd isotropy of the early Universe
),
R
(
t
)
is the same over all space and
d
epends only time. The value o
f
R
(
t
)
at tim
e
t
is re
l
ate
d
to its va
l
ue to
d
a
y
at
t
=
t
0
v
i
aa
r
eds
hi
ft facto
r:
R
(
t
)=
R
(
t
0
)
/
(
1
+
z
)
. Hubble’s law on the
ex
p
ansion of the Universe can then be ex
p
ressed as
˙
R
(
t
)
=
HR
(
t
)
,
(
9.1
)
whe
r
e
˙
R
≡
d
R
d
t
.O
ne may also de
fi
ne a dimensionless scale paramete
r
a
(
t
)
≡
R
(
t
)
/R
(
t
0
) to express Hubble’s law: ˙
a
(
t
)
=
Ha
(
t
).
T
he evolution of the Universe is governed by the solution of Einstein’
s
field equations of general relativity. Given a homogeneous and isotropic dis-
tribution o
f
matter and radiation behavin
g
like the per
f
ect
f
rictionless
fl
uid
,
E
instein’s equations lead to the Friedmann equations
(
Friedmann, 1922
):
H
2
=
˙
R
R
2
=
8
π
G
N
ρ
3
−
k
R
2
+
Λ
3
,
(
9.2
)
H
2
+
˙
H
=
¨
R
R
=
−
4
π
G
N
3
(
ρ
+
3
p
)
+
Λ
3
,
(
9.3
)
whe
r
e
G
N
i
sNewton
’
s
g
ravitationa
l
constant
,
ρ
d
enotes t
h
eener
g
y
d
ensity
,
p
r
e
p
resents the isotro
p
ic
p
ressure,
k
i
s the curvature constant
,
an
d
Λ
is the
c
osmological constant. Differentiating Eq.
(
9.2
)
,weobtai
n
˙
H
=
4
π
G
N
3
H
˙
ρ
+
k
R
2
(
9.4
)
with the help of Eq.
(
9.1
)
. A combination of Eqs.
(
9.2
)
,
(
9.3
)
and
(
9.4
)
yield
s
1
N
ote t
h
at 1 Mpc = 10
3
k
pc = 10
6
p
c, w
h
ere pc = parsec is a common
l
y-use
d
unit o
f
cosmic distances. 1
p
c=3
.
085
7
×
10
1
3
k
misde
fi
ned as the distance tha
t
an object has to be for its parallax to equal 1 second of arc (or 1/3600 of a degree).
9
.1 Neutrinos in t
h
eEar
l
y Universe 325
˙
ρ
=
−
3
H
(
ρ
+
p
)
.
(
9.5
)
In some literature the so-called Friedmann equation only refers to Eq.
(
9.2
).
T
he ultimate fate of the Universe is determined b
y
the curvature constan
t
k
,
w
h
ic
h
can on
l
yta
k
et
h
e
d
iscrete va
l
ues 0 an
d
±
1. In astrop
h
ysics
k
=+
1,
k
=
−
1a
n
d
k
= 0 correspond respectivel
y
to closed, open and spatiall
yfl
a
t
geometries. Let us assum
e
Λ
=
0inEq.
(
9.2
)
to discuss the cosmological
m
eaning o
f
k
.
T
he
k
= +1 case correspon
d
stoac
l
ose
d
Universe w
h
ic
h
expands and reaches a maximum radius and then recollapses in a
fi
nite time
,
whereas th
e
k
=
−
1 case corres
p
onds to an o
p
en Universe which ex
p
ands
wit
h
out any
l
imit. Fo
r
k
= 0, the Universe is flat in the sense that it expand
s
f
orever but the velocity tends asymptotically to zero at su
ffi
ciently lar
ge
t
. But these simple observations may change i
f
Λ
i
s nonvanishing. Current
observational data indicate that the Universe is extremely close to a
fl
a
t
U
niverse
(
i.e.
,
k
≈
0
)
on large scales
.
9.1.2 The Ener
g
yDensityo
f
the Univers
e
T
h
eener
g
y
d
ensity
ρ
an
d
t
h
e isotropic pressure
p
i
nEq.
(
9.3
)
are in general
related to each other through the equation of state
:
p
=
w
ρ
,
where
w
i
sa
d
imensionless parameter. I
f
w
i
s time-independent, then Eq.
(
9.5
)
become
s
˙
ρ
=
−
3(
1+
w
)
ρ
˙
R
/R
a
n
d
can
b
e easi
l
yinte
g
rate
d
.T
h
e resu
l
tis
ρ
∝
R
−
3
(
1
+
w
)
.
(
9.6
)
The total ener
g
y density o
f
the Universe may consist o
f
three components
:
ρ
=
ρ
m
+
ρ
r
+
ρ
v
,
(
9.7
)
whe
r
e
ρ
m
,
ρ
r
a
n
d
ρ
v
stand respectively
f
or the ener
g
y densities o
f
matter
,
radiation and vacuum. The role o
f
ρ
v
i
s equivalent to that of the cosmologica
l
co
n
sta
n
t
Λ
i
nt
h
eFrie
d
mann equation; i.e.
,
Λ
=
8
π
G
N
ρ
v
.Eq.
(
9.2
)
can then
be
r
ew
ri
tte
n
as
H
2
=
˙
R
R
2
=
8
π
G
N
3
(
ρ
m
+
ρ
r
+
ρ
v
)
−
k
R
2
.
(
9.8
)
In the following we discuss the simplified solutions to Eq.
(
9.8
)
by assuming
t
h
eener
g
y
d
ensity
ρ
to be dominated b
y
one o
f
its three components, an
d
s
ummarize relevant results in Table 9.1
.
(
1
)
A radiation-dominated
U
nivers
e
.
T
h
eear
l
y Universe was very
h
ot an
d
d
ense, and thus its energy density was dominated by radiation
(
or relativistic
particles
)
. A gas of radiation can be described by the equation of state
p
=
ρ
r
/
3(
i.e.
,
w
=1
/
3)
.InthiscaseEq.
(
9.6
)
become
s
ρ
r
∝
R
−
4
.
Ne
gl
ectin
g
t
h
e
te
rm
s assoc
i
ated w
i
t
h
ρ
m
,
ρ
v
a
n
d
k
in Eq.
(
9.8
)
,weimmediatelyarriveat
˙
R
∝
1
/
R. As a result
,
R
(
t
)
∝
t
1
/
2
a
nd
H
=
1
/
(
2
t
)
.