Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
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346 9 Big Bang Nuc
l
eosynt
h
esis an
d
Re
l
ic Neutrinos
I
f
neutrinos are Dirac particles, which can have vectorial couplin
g
s, this tiny
a
cceleration is described by
(
Ferreras and Wasserman, 1995; Duda et al.
,
2
001; Gelmini, 2005; Ringwald, 2009
)
8
a
≈
ν,
¯
ν
4
π
3
N
2
A
N
ρ
r
3
σ
νN
(
n
ν
v
re
l
)
:
;<
;
=
=
fl
u
x
(2
m
ν
v
r
el
)
:
;<
;
=
=
mo
m
.
t
ransfer
≈
2
×
10
−
28
c
m
s
−
2
n
ν
¯
n
ν
10
−
3
c
v
r
el
ρ
g
c
m
−
3
r
λ
ν
3
,
(
9.58
)
whe
r
e
N
A
N
is Avo
g
adro’s number
,
σ
νN
≈
G
2
F
m
2
ν
/
π
m
easu
r
es the c
r
oss sec
-
tion o
f
elastic neutrino-nucleon scatterin
g
,an
d
v
r
el
≡|
v
−
v
⊕
|
de
n
otes t
h
e
m
ean velocity of the relic neutrinos in the rest system of the detector wit
h
|
v
⊕
|≈
7
.
7
×
10
−
4
c
b
ein
g
the velocity o
f
the Earth throu
g
h the Milky Way. I
f
n
eutrinos are Majorana particles, which only have axial couplin
g
s, the corre-
s
ponding acceleration involves a further suppression factor
(
v
r
el
/c
)
2
∼
10
−
6
f
or an unpolarized tar
g
et or
v
r
el
/c
∼
10
−
3
f
or a polarized target
(
Hagmann
,
1
999
)
. The minimal acceleration that can be detected by current Cavendish
-
type torsion balances is o
f
O
(
1
0
−
13
)
cm
/
s
2
(
Adelberge
r
et al
.
,
2009
)
,over
-
whelmin
g
ly lar
g
er than the expected si
g
nal.
S
olar neutrinos and dark matter
a
re likely to produce a much larger background
(
Strumia and Vissani, 2006
)
.
(
3
)
T
he Stodolsky effec
t
.
This effect consists of an energy split of the tw
o
s
pin states o
f
non-relativistic electrons in the
C
ν
B; i.e., the ener
g
yo
f
an elec-
tron receives an extra contributio
n
Δ
E
e
∼
G
F
s
·
v
(
n
ν
−
n
ν
)
that relies on th
e
d
irection o
f
its spin
s
w
it
h
respect to t
h
ere
l
ic neutrino win
d
v
(
Stodolsky
,
1
974
)
. The magnitude of
Δ
E
e
d
e
p
ends on
G
F
i
n
stead of
G
2
F
, but it mi
g
h
t
be suppressed by the asymmetry of relic neutrino and antineutrino number
de
n
s
i
t
i
es
.Th
e
l
atte
ri
sdefi
n
ed as
η
ν
≡
(
n
ν
−
n
ν
)
/
n
γ
.The
S
todolsky e
ff
ect i
s
a
lso dependent on the Dirac or Ma
j
orana nature o
f
non-relativistic neutrino
s
(
Gelmini, 2005
)
. It practically manifests itself as a torqu
e
T∼
N
e
N
Δ
E
e
ac
t
-
i
n
g
on a ma
g
netize
d
macroscopic o
b
ject wit
h
N
e
N
po
l
arize
d
e
l
ectrons. In t
h
e
t
y
pical case of one polarized electron per atom, an ob
j
ect with a linear siz
e
r an
d
atomic num
b
e
r
A
m
ay experience a tiny acce
l
eration in
d
uce
db
yt
he
relic neutrino wind
(
Strumia and Vissani, 2006
):
a
∼
T
AN
e
N
m
e
r
∼
10
−
28
c
m
s
−
2
100
A
c
m
r
v
10
−
3
η
ν
.
(
9.59
)
We see that the estimates in Eqs.
(
9.58
)
and
(
9.59
)
are comparable in mag
-
n
itude, far below the
p
resent ex
p
erimental sensitivities
.
(
4
)
Z
-bursts of relic and UHE neutrino
s
. As discussed in Section 8.4.1
,
u
ltrahigh-energy
(
UHE
)
cosmic neutrinos may interact with the C
ν
Ba
n
d
get absorbed via the following reactions taking place on the
Z
r
eso
n
a
n
ce:
8
S
ince relic neutrinos are mostl
y
non-relativistic toda
y
, their e
ff
ects in question
are significantly dependent on their Dirac or Majorana nature (Gelmini, 2005)
.
R
e
f
erences 34
7
ν
UH
E
+
ν
C
ν
B
→
Z
→
had
r
o
n
sa
n
d
ν
U
H
E
+
ν
C
ν
B
→
Z
→
h
adrons
(
Weiler,
1
982
)
.Insucha
Z
-burst event the kinetic ener
g
y of an UHE neutrin
o
ν
i
o
u
g
ht
to
be
E
ν
i
≈
M
2
Z
M
/
(2
m
i
)
≈
4
.
2
×
(
e
V
/m
i
)
×
10
21
eV
,
w
h
ere
m
i
s
tands for th
e
mass of
ν
i
.O
ne ma
y
use the
Z
-burst mechanism to probe the existence o
f
the
C
ν
B
in two wa
y
s: one is to detect the “emission” feature of
Z
-
burst
s
c
haracterized by a directional excess of UHE cosmic rays, gamma rays an
d
n
eutrinos above the well-known
G
ZK cuto
ff
E
GZK
∼
5
×
10
19
e
V
(
see Sectio
n
8
.1.1 for a detailed discussion
)
, and the other is to observe the “absorption
”
f
eature in the UHE cosmic neutrino flux which is just the consequence of its
i
nteractions with the
C
ν
B
o
n
the
Z
resonance
(
Weiler 1982; Roulet, 1993;
Yoshida, 1994
)
. But in either case the signal rate is too small to be detecte
d
by means o
f
the present technolo
g
yo
f
UHE neutrino telescopes.
R
e
f
erence
s
A
delber
g
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g
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ennow, M., 2008, P
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roadhurst, D., 2010, arXiv:1004.4238
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occo,
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occo, A. G., Mangano, G., and Messina, M., 2009, Phys. Rev.
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uda,
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elmini,
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ru
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agmann, C., 1999, arXiv:astro-ph/9902102
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annestad,
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u, W.,
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e
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k
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d
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l
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e
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l
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g
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g
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l
eosynt
h
esis an
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Re
l
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i, Y. F., Xing, Z. Z., and Luo, S., 2010, Phys. Lett.
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an
g
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l
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., and Skillman, E., 2004,
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ert, M.,
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in
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einber
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oshida, S., 1994, Astropart. Phys. 2
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10
N
eutrinos and
C
osmological
S
tructures
A
s fairly stable and weakly interactin
g
particles, relic neutrinos of the Bi
g
Bang must survive today and form a cosmic background similar to the cosmi
c
m
icrowave background
(
CMB
)
radiation. This cosmic neutrino background
(C
ν
B
)
played an important role in the evolution of the Universe. A measure
-
m
ent of the CMB anisotropies can tell us a lot of cosmological information
,
i
ncludin
g
the number o
f
neutrino species and neutrino masses.
S
ince mas
-
s
ive neutrinos became non-relativistic when the tem
p
erature fell below thei
r
m
asses, they should have left an imprint on the clustering of galaxies. In this
c
hapter we shall first describe how the CMB and large-scale structures
(
LSS
)
f
ormed
,
and then discuss how to constrain neutrino masses from the mea
-
s
urements of the
C
MB and L
SS
.
W
e shall also comment on sterile neutrinos
a
sa
p
ossible candidate o
f
warm dark matter in the Universe
.
10.1 The Cosmic Microwave Back
g
roun
d
The CMB radiation was first discovered b
yA
rno Penzias and Robert Wilso
n
i
n 1965
(
Penzias and Wilson, 1965
)
. The significance of this discovery was
reco
g
nized with the Nobel Prize in Physics in 1978.
S
ince then the
C
M
B
h
as become one of the most important pillars of the Big Bang model of
c
osmolo
g
y. In 2006, John Mather and
G
eor
g
e
S
moot received the Nobel Priz
e
i
nPh
y
sics
f
or their discover
y
o
f
the blackbod
yf
orm and anisotrop
y
o
f
the
CMB radiation by means of the COBE satellite
(
Smoo
t
e
tal., 1992
)
.I
t
turns out that the
C
MB anisotropy measurements can place very precis
e
c
onstraints on a number o
f
cosmolo
g
ical parameters, includin
g
the sum o
f
n
eutrino masses, and thus have led us to the era of precision cosmology.
10.1.1 Matter-radiation Equalit
y
A
s discussed in Section 9.2, the primordial nucleos
y
nthesis started at abou
t
t
≈
2 s after the Big Bang. It was completed by abou
t
t
≈
1
0
3
s
,
after the tem-
Z.-Z. Xing et al., Neutrinos in Particle Physics, Astronomy and Cosmology
© Zhejiang University Press, Hangzhou and Springer-Verlag Berlin Heidelberg 2011
350 10 Neutrinos and
C
osmological
S
tructures
perature had
f
allen to the level o
f
O
(
1
0
−
2
)
MeV
(
Grupe
n
et al
., 2005
)
.Any
n
eutrons that had not become bound into heavier nuclei b
y
this time could no
l
onger survive but would soon decay. Since then the early Universe evolved
i
nasomewhatdi
ff
erent wa
y
. The next remarkable event took place whe
n
the energy density of radiation
(
i.e., relativistic particles, including photon
s
a
nd neutrinos
)
became as low as the energy density of matter
(
i.e., non
-
relativistic particles, including nuclei and electrons
)
. This matter-radiatio
n
equality signifies a transition of the Universe from its radiation-dominated
era to its matter-dominated epoch. In the following let us work out the tim
e
o
f
matter-radiation equalit
y
.
T
o estimate when the matter-radiation equality happened in the evolution
o
f
the early Universe, one needs some in
f
ormation on the ener
g
y densities
o
f
matter and radiation. It will be seen that a
g
lobal analysis o
f
current
observational data on the CMB and LSS yield
s
Ω
m
(
t
0
)
≈
0
.
26
for non-
r
elat
i
v
i
st
i
c
m
atte
r
a
n
d
Ω
v
(
t
0
)
≈
0
.
74
f
or vacuum ener
g
yi
f
a
fl
at Univers
e
w
i
th
k
= 0 is assumed toda
y
a
t
t
=
t
0
(
Komats
u
et al
.
,
2009; Nakamur
a
et
al.
,
2010
)
. In comparison, the energy density of photons ρ
r
h
as a contri
b
u
-
tion
Ω
r
(
t
0
)
≈
4
.
84
×
10
−
5
as estimated below Eq.
(
9.25
)
. Because the energ
y
d
ensit
y
of relativistic neutrinos
ρ
ν
i
s related to
ρ
r
throu
g
h
ρ
ν
ρ
r
=
7
8
·
g
ν
g
γ
T
ν
TT
T
γ
TT
4
=
21
8
4
11
4
/
3
≈
0
.
681
,
(
10.1
)
we obta
i
n
Ω
ν
(
t
0
)
=
(
ρ
ν
/
ρ
γ
)
Ω
r
(
t
0
)
≈
3
.
30
×
10
−
5
fo
rn
eut
rin
os
1
.T
he sum
Ω
r
(
t
0
)
+
Ω
ν
(
t
0
)
≈
8
.
14
×
10
−
5
t
urns out to be about 3200 times smalle
r
t
h
a
n
Ω
m
(
t
0
)
. In addition, the energy density of non-relativistic baryons has a
co
n
t
ri
but
i
on
Ω
B
(
t
0
)
≈
0
.
043 as given below Eq.
(
9.51
)
. Table 9.1 has provide
d
u
s with the proportionalit
y
relation
s
Ω
r
∝
ρ
r
∝ 1
/R
4
and
Ω
m
∝
ρ
m
∝
1
/R
3
.
Taking account of
R
(
t
)=
R
(
t
0
)
/
(
1
+
z
)
,wefin
d
Ω
r
(
t
)
Ω
m
(
t
)
=
Ω
r
(
t
0
)
R
4
(
t
0
)
R
4
(
t
)
·
R
3
(
t
)
Ω
m
(
t
0
)
R
3
(
t
0
)
=(
1+
z
)
Ω
r
(
t
0
)
Ω
m
(
t
0
)
.
(
10.2
)
Th
e
n
t
h
e
r
eds
hi
ft
z
fo
r
t
h
e
m
atter-p
h
oton
e
qua
l
it
y
Ω
r
(
t
)=
Ω
m
(
t
)
is given by
z
=
Ω
m
(
t
0
)
Ω
r
(
t
0
)
−
1
≈
537
2
.
(
10.3
)
In a similar wa
y
, the redshi
ft
z
fo
r
t
h
e
b
ar
y
on-photo
n
equalit
y
Ω
r
(
t
)
=
Ω
B
(
t
)
i
s found to be
z
=
Ω
B
(
t
0
)
Ω
r
(
t
0
)
−
1
≈
887 ;
(
10.4
)
1
N
ote that we have assumed neutrinos to be relativistic particles. When their
masses are taken into account, at least two mass eigenstates of the relic neutrinos
are a
l
rea
dy
non-re
l
ativistic to
d
a
y
.Int
h
is cas
e
Ω
ν
(
t
0
)
should be estimated in a
different way, as discussed in Section 10.2.
1
0.1 The
C
osmic Microwave Background 35
1
a
n
dt
h
e
r
eds
hi
ft
z
f
or the equalit
y
Ω
r
(
t
)
+
Ω
ν
(
t
)
=
Ω
m
(
t
)
between the energy
d
ensit
y
of matte
r
a
nd that o
f
all relativistic
p
articles
(
i.e., both photons and
n
eutrinos
)
is given b
y
z
=
Ω
m
(
t
0
)
Ω
r
(
t
0
)+
Ω
ν
(
t
0
)
−
1
≈
3193
.
(
10.5
)
In order to
fig
ure out when the above equalities took place, one needs t
o
establish the relations between
t
an
d
t
0
via the redshift
z
.
For illustration
,
we simp
l
y assume t
h
at t
h
eUniverse
h
as
b
een matter-
d
ominate
d
since t
h
e
m
atter-ra
d
iation equa
l
it
y
.Int
h
is case
t
∝
R
3
/
2
h
o
ld
sass
h
own in Ta
bl
e9.1,
a
nd therefor
e
t
is related t
o
t
0
through
t
=
t
0
R
(
t
)
R
(
t
0
)
3
/
2
=
t
0
(
1
+
z
)
3
/
2
,
(
10.6
)
whe
r
e
t
0
≈
13
.
7 Gyr is the age of the Universe
(
Komats
u
et al
., 2009
)
.Then
we obtai
n
t
mr
≈
3
.
5
×
10
4
y
rfo
r
z
≈
5372 or
t
mr
≈
7
.
6
×
1
0
4
y
rforz
≈
3
193.
N
ote that these numbers can only give us a ballpark feeling of the magnitude
of
t
mr
,
b
ecause we
k
now t
h
at t
h
e vacuum ener
g
yma
k
es up an important
portion of the total energy density of the Universe
(
i.e.
,
Ω
v
(
t
0
)
≈
0
.
74 today
)
a
nd hence the assumption of matter dominance is more or less problematic
.
W
hen the energy density of matter exceeded that of radiation
(
i.e., fo
r
z
<
3
000 as indicated by Eq.
(
10.5
))
, the gravitational clustering of matter could
ta
k
ep
l
ace. Dar
k
matter was vita
ll
yimportant
h
ere,
b
ecause t
h
e
d
ominanc
e
of baryons alone over radiation would not occur until very much later
(
e.g.
,
at
z<
9
00 as indicated by Eq.
(
10.4
)
or after the decoupling of photons fro
m
m
atter
)(
Perkins, 2009
).
1
0
.1.2 Formation of the
C
MB
In the radiation-dominated epoch any protons and electrons that managed
to
b
in
d
into neutra
lh
y
d
rogen atoms via t
h
e reactio
n
p
+
e
−
→
H
+
γ
would
b
e imme
d
iate
ly
p
h
oto
d
issociate
d
via t
h
e inverse reaction H
+
γ
→
p
+
e
−
.As
the tem
p
eratur
e
T
d
ropped significantly below the electron binding energy of
h
ydrogen
(
i.e.,
E
b
in
d
=13
.
6
eV
)
, the formation of hydrogen became feasible
a
nd the Universe trans
f
ormed
f
rom an ionized plasma into a
g
as o
f
neutra
l
a
toms. This process is usually referred to a
s
r
ecombination
.W
hen the number
d
ensity o
ff
reeelectronsinthecosmoswassu
ffi
ciently small, the mean
f
ree
path o
f
a photon would be su
ffi
ciently lon
g
such that most photons have
n
ot scattered with matter since then. This phenomenon is usually called th
e
d
ecouplin
g
o
f
photons
f
rom matter
.
Let us estimate the tem
p
erature and time of recombination. The
p
roces
s
p
+
e
−
↔
H
+
γ
sh
ou
ld b
eint
h
erma
l
equi
l
i
b
rium w
h
en t
h
etemperatur
e
T
was
a
bove the ionization ener
g
yo
f
hydro
g
e
n
E
b
in
d
=13
.
6
e
V
.
W
e are intereste
d
352 10 Neutrinos and
C
osmological
S
tructures
i
n what happened as the temperature fell.
A
naive expectation is that a
s
i
g
nificant amount of hydro
g
en could have been formed when
T
d
ro
pp
ed
b
e
l
ow E
b
in
d
. But the smallness of the baryon-to-photon rati
o
η
≡
n
B
/
n
γ
∼
6
×
10
−
10
fixed during the Big Bang nucleosynthesis
(
BBN
)
implies that the
n
umber of
p
hotons wit
h
E
>
E
b
in
d
m
i
g
ht be comparable with the number of
baryons only at a much lower temperature. Note that the rate of the reactio
n
p
+
e
−
→
H
+
γ
i
s proportional to the product o
f
the number densities o
f
p
rotons and electrons
n
p
n
e
,
and the rate of the reaction H + γ
→
p
+
e
−
i
s
proportional to the number density of hydrogen atoms
n
H
2
.
According t
o
the Saha equation
(
Saha, 1921
)
,
n
p
n
e
n
H
=
m
e
T
2
π
3
/
2
exp
−
E
b
ind
T
.
(
10.7
)
H
ere t
h
e tota
lb
ar
y
on num
b
er
d
ensit
y
i
s
n
B
=
n
p
+
n
H
.
Usin
g
x
to de
n
ote
the fraction of hydrogen atoms which are ionized, we hav
e
n
e
=
n
p
=
x
n
B
a
n
d
n
H
=
(
1
−
x
)
n
B
.
Then Eq.
(
10.7
)
can be rewritten a
s
x
2
n
B
1
−
x
=
m
e
T
2
π
3
/
2
e
x
p
−
E
bind
T
.
(
10.8
)
O
ne may numerically illustrate the chan
g
es o
f
x
w
i
th
T
b
y inputtin
g
t
he
values o
f
m
e
an
d
E
bind
. It is found tha
t
x
c
atastrophicall
y
drops fo
r
T
to vary from 0.35 eV to 0.25 eV
(
Perkins, 2009
)
. This result implies tha
t
the Universe was trans
f
ormin
gf
rom an ionized plasma into an essentiall
y
n
eutral gas of hydrogen
(
and helium
)
at the recombination temperature
T
rec
TT
≈
0
.
3 eV. A comparison betwee
n
T
rec
TT
and today’s CMB temperatur
e
T
0
T
T
=
2
.
7
25K=2
.
3
4
8
×
10
−
4
eV a
ll
ows us to
d
etermine w
h
en recom
b
ination
took
p
lace. Because of
T
∝ 1
/
R
,
the redshift at the time of recombination is
1+
z
r
ec
=
R
(
t
0
)
R
(
t
re
c
)
=
T
rec
T
T
T
0
T
T
.
(
10.9
)
T
husweobta
i
n
z
rec
≈ 1277. Assuming the Universe to be matter-dominated
f
rom this point up to toda
y
,wehav
e
t
∝
R
3
/
2
∝
T
−
3
/
2
a
n
dt
h
us
t
r
ec
=
t
0
T
0
TT
T
rec
TT
3
/
2
=
t
0
(
1
+
z
r
ec
)
3
/
2
.
(
10.10
)
G
ive
n
t
0
≈
13
.
7
GyrastheageoftheUniverse,thetimeofrecombinationis
t
h
e
n
fou
n
dtobe
t
r
ec
≈
3
.
0
×
10
5
yr
.
S
hortl
y
after recombination, the mean free path of a photon became so
l
ong that radiation and matter were effectively decoupled. The decouplin
g
2
S
ince the number densit
y
o
f
photon
s
n
γ
is remar
k
a
bl
y
l
ar
g
er t
h
a
n
n
p
,
n
e
a
n
d
n
H
, the reaction rate is essentially insensitive t
o
n
γ
(
Perkins, 2009).
1
0.1 The
C
osmic Microwave Background 35
3
o
f
photons
f
rom matter was o
f
course not an instantaneous process, becaus
e
a
photonemitteduponrecombinationofonehydro
g
en atom could almost
i
mme
d
iate
l
y ionize anot
h
er
h
y
d
rogen atom a
t
T
∼
T
rec
TT
.
Wh
i
l
et
h
e
U
niverse
was an ionize
d
p
l
asma
d
urin
g
t
h
is perio
d
,t
h
ep
h
oton scatterin
g
cross sec
-
tion was dominated by the Thomson scattering
(
i.e., the elastic scattering
of electromagnetic radiation by a free charged particle
)
. The mean free pat
h
o
f
a photon was there
f
ore determined by the decreasin
g
number density o
f
electrons as the Universe expanded. This path length became longer than th
e
h
orizon distance
(
i.e., the radius of the observable Universe at a given time
)
a
tt
h
e
d
ecoup
l
in
g
temperature
T
dec
TT
≈
0
.
26 eV, correspondin
g
to a redshi
f
t
z
d
e
c
≈
1
100
(
Grupe
n
e
tal
.
,
2005
)
. The decoupling time is determined by
t
d
e
c
=
t
0
T
0
TT
T
dec
T
3
/
2
=
t
0
(
1+z
dec
)
3
/
2
.
(
10.11
)
S
owe
g
e
t
t
dec
≈
3
.
8
×
10
5
y
r, the approximate moment
f
or the
C
MB to
f
orm
.
A
fter the decoupling of radiation from matter, the latter became trans
-
parent to the former and thus the formation of atoms and molecules could
be
g
in in earnest. We may de
fi
ne the “sur
f
ace o
f
last scatterin
g
” as the spher
e
c
entered about us with a radius e
q
ual to the mean distance to the last
p
lac
e
where the CMB photons scattered
(
Grupe
n
et al
.
,
2005
)
. Such a distance
i
s approximately equal to the one to where the matter-radiation decouplin
g
took place, and the time of last scattering is essentially the same a
s
t
d
e
c
.I
n
this sense, today’s detection o
f
the
C
MB is actually probin
g
the condition
s
i
n the Universe at a time o
f
rou
g
hly
3
.
8
×
1
0
5
years a
f
ter the Bi
g
Ban
g.
10.1.3 Anisotro
p
ies o
f
the CM
B
The spectrum o
f
the
C
MB measured initiall
y
b
y
Penzias and Wilson coul
d
well be described b
y
ablackbod
y
function with a direction-independent tem-
pera
t
ur
e
T
∼
3 K. In the 1970’s it was found that the temperature was a
bit hi
g
her in one particular direction o
f
the microwave sky than in the op-
posite direction. This interestin
g
anisotropy of the CMB, of
O
(
10
−
3
)
,wa
s
i
nterprete
d
as
b
eing cause
db
yt
h
e Eart
h’
smotionw
h
ic
h
s
h
ou
ld b
e equiva-
l
ent to a peculiar velocity of the Milky Way
(
i.e.
,
v
MW
∼
630
k
m
s
−
1
)
.I
n
1
992, the COBE satellite discovered the tem
p
erature variations o
f
O
(
1
0
−
5
)
i
n
the CMB at smaller angular separations
(
Smoot
et al
.
,
1992
)
. These remark
-
a
ble
C
MB anisotropies have been con
fi
rmed by a number o
fg
round-based
a
nd ballon-borne measurements with much better sensitivities and angular
resolutions
(
Whit
e
et al.
, 1994; Hu and Dodelson, 2002
)
. In particular, th
e
WM
A
P satellite has measured the CMB and its anisotro
p
ies to an un
p
rece
-
d
entedly good degree of accuracy since 2003
(
Bennet
t
e
tal.
,
2003
;
Jarosik
et
al.
,
2007; Hins
h
a
w
et al
., 2009
).
T
o describe the
C
MB anisotropies over a wide ran
g
eo
f
an
g
ular scales
,
one may consider the CMB temperature as a direction-dependent functio
n
354 10 Neutrinos and
C
osmological
S
tructures
T
(
θ
,
φ
)
,where
θ
a
n
d
φ
s
tand respectively
f
or the polar and azimuthal an
g
les
i
n the spherical coordinate s
y
stem. This function can be expanded in terms
of the spherical harmonic function
s
Y
m
YY
(
θ
,
φ
)
as follows:
T
(
θ, φ
)=
∞
=0
m
=
−
a
m
Y
m
Y
Y
(
θ, φ
)
.
(
10.12
)
Such an expansion of the CMB sky is analogous to a Fourier series, in which
the hi
g
her-order terms correspond to hi
g
her
f
requencies. Here the terms wit
h
h
i
g
her
values point to the structures o
f
the
C
MB at smaller an
g
ular scales
.
A
term in the series of Eq.
(
10.12
)
is usually referred to as a multipole mo
-
me
n
t
:
the
=0termist
h
e monopo
l
e; t
he
=
1termist
h
e
d
ipo
l
e; an
d
s
o
on. This lan
g
ua
g
e is borrowed from the multipole expansion used frequently
i
n the study of electromagnetic and gravitational fields. Current theoretica
l
m
o
d
e
l
s
g
enera
ll
ypre
d
ict t
h
at t
h
e
a
m
m
odes are
G
aussian random
fi
elds t
o
ag
ood de
g
ree of precision, and possible non-Gaussian effects are expecte
d
to be one or two orders of magnitude smaller than the present observationa
l
l
imits
(
Bartol
o
et al.
,
2004
)
. A statistically isotropic CMB sky means that
a
ll m
v
alues are e
q
uivalent; i.e., there is no
p
referred axis. Given this ob
-
s
ervation and the Gaussian statistics, the variance of the temperature fiel
d
(
or equivalently, the power spectrum i
n
)
can fully characterize the CM
B
a
nisotropies
(
Nakamura et al.
,
2010
)
. The power summed over all possibl
e
m
values at each
is
(2
+1
)
C
C
/
(4
π
)
,wher
e
C
C
≡
|
a
m
|
2
is defi
n
ed
.Th
e set
of
C
C
(
for
=0
,
1
,
···
,
∞
)
is called the angular power spectrum. The valu
e
of
C
C
r
epresents the level of the CMB structure measured at an angular sep-
a
r
at
i
on
Δ
θ
≈
π
/
(
Grupen
et al
.
,
2005
)
. A measurement can only be able t
o
resolve an
g
les down to a minimum value,
f
rom which the maximal value o
f
i
s then deducible
.
(
1
)
Th
e monopo
l
e.T
he
=
0 term in the Laplace expansion o
f
T
(
θ
,
φ
),
which can be re
g
arded as the monopole component of the CMB spectrum
,
s
ignifies the temperature of the CMB averaged over all directions of the sk
y
a
sshowninthele
f
t panel o
f
Fi
g
. 10.1. Its value with the
1
σ
e
rr
o
r
ba
ri
s
T
γ
TT
=2
.
7
2
5
±
0
.
001 K today
(
Mather
e
tal
.
, 1999
)
. The monopole distributio
n
of the CMB is consistent with a blackbody function wit
h
T
=
T
γ
TT
,
an
d
t
he
c
orrespondin
g
number density o
f
relic photons is known as
n
γ
≈
4
11
c
m
−
3
.
(
2
)
The di
p
ole
.
The largest anisotropy of the CMB, with an amplitude of
Δ
T
=
3
.
355
±
0
.
0
08 mK
(
Hinshaw
et al
.
,
2009
)
and a “yin-yang” pattern as
s
hown in the middle panel o
f
Fi
g
. 10.1, is described by th
e
=
1
te
rm in
the
L
a
p
lace ex
p
ansion of
T
(
θ
,
φ
)
. This dipole component of the CMB spectrum
i
sc
h
aracterize
db
yt
h
e temperature variatio
n
Δ
T/T
γ
TT
≈
1
.
23
×
10
−
3
a
n
d
a
ttributed to the Doppler shi
f
t caused b
y
the Earth’s motion relative to the
n
early isotropic blackbody field, as confirmed by a measurement of the radia
l
velocity of local galaxies
(
Courtea
u
et al.
,
2000
)
. To be specific, the motio
n
o
f
an observer in the solar s
y
stem with velocit
y
v
with res
p
ect to the rest
f
rame of an isotro
p
ic Planckian radiation field of tem
p
eratur
e
T
0
TT
p
roduces
a
1
0.1 The
C
osmic Microwave Background 35
5
Fig. 10.1 The monopole
(
left
)
,dipole
(
middle
)
and multipole
(
right
)
patterns in
the cosmographic map of the CMB measured by the WMAP satellite
(
Courtesy o
f
the WMAP Science Team, http:
//
map.gsfc.nasa.gov
/)
D
oppler-shifted temperature pattern
(
Peebles and Wilkinson, 1968
)
T
(
θ
)
=
T
0
TT
1
−
β
2
1
−
β
cos
θ
≈
T
0
T
T
1+
β
cos
θ
+
O
(
β
2
)
,
(
10.13
)
whe
r
e
θ
i
sthean
g
le between the line o
f
si
g
ht and the direction o
f
the motio
n
of the observer
,
an
d
β
≡
v/
c
w
ith c
b
ein
g
the speed of li
g
ht. The observe
d
s
pectrum of the CMB as a function o
f
θ
is given by
(
Fixse
n
et al
., 1994
)
S
ν
(
θ
)=
B
ν
(
T
)
≈
B
ν
(
T
0
T
T
)+
T
0
T
T
β
cos
θ
d
B
ν
d
T
T
=
T
0
TT
,
(
10.14
)
w
h
e
r
e
B
ν
(
T
)
denotes the Planck function. One observes a blackbody spec
-
trum of the CMB at every point in the microwave sky with temperatur
e
T
(
θ
)
. The monopole part of the spectrum is just described by
B
ν
(
T
0
TT
)
, while
the di
p
ole
p
art o
f
the s
p
ectrum must have the sha
p
eo
f
the derivative o
f
B
ν
(
T
)
evaluated at the temperature T
=
T
0
TT
.G
ive
n
T
0
TT
=
T
γ
T
T
together wit
h
Δ
T
=3
.
355
±
0
.
008 mK, the implied velocity
f
or the solar system barycenter i
s
v
=
369
.
0
±
0
.
9
km s
−
1
towards
(
, b
)
=
(
263
.
9
9
◦
±
0
.
14
◦
,
48
.
2
6
◦
±
0
.
0
3
◦
)(
Hin
-
sh
aw
e
ta
l
.
,
2009
)
. The dipole anisotropy means that the CMB is blueshifted
to a sli
g
htly hi
g
her temperature in the direction o
f
motion and redshi
f
ted in
the opposite direction. Because the dipole is a frame-dependent quantit
y
,on
e
m
ay define a “local rest frame” of the Universe in which the CMB has n
o
d
ipole anisotropy at all
(
Grupe
n
et al
.
,
2005
)
.
(
3
)
T
he multi
p
oles
.
The
2
terms in the La
p
lace ex
p
ansion of
T
(
θ
,
φ
)
c
onstitute the multipole part or small-angle anisotropies of the CMB spec
-
trum c
h
aracterize
dby
t
h
e temperature variation
s
Δ
T/T
γ
T
T
∼
10
−
5
.
I
t was the
COBE satellite that first discovered such small-angle CMB anisotropies, be
-
c
ause it
h
a
d
an angu
l
ar reso
l
ution
Δ
θ
∼
7
◦
a
n
d
cou
ld
pro
b
et
h
eangu
l
ar power
sp
ectrum u
p
toamulti
p
ole number
≈
π
/
Δ
θ
∼
20
(
Smoo
t
et al
., 1992
)
.An
i
mpressive multipole pattern in the cosmographic map of the CMB, as show
n
i
n the right panel of Fig. 10.1
(
with the dipole component subtracted
)
,ha
s
been established from the measurements of the WM
A
P satellite which has
a
n angular resolutio
n
Δ
θ
∼
0
.
2
◦
a
nd can therefore determine the angular