Xing Zh., Zhou Sh. Neutrinos in Particle Physics, Astronomy and Cosmology
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356 10 Neutrinos and
C
osmological
S
tructures
power spectrum up to
∼
1
000
(
Bennett
et al
.
,
2003; Jarosi
k
et al
.
,
200
7;
H
insha
w
et al.
,
2009
)
. The observed multipole effects can be interpreted a
s
being mostly the result of small density fluctuations in the early Universe,
m
ani
f
estin
g
themselves in the epoch o
f
last scatterin
g
o
f
the
C
MB photons
.
Such small density fluctuations could be amplified by gravity, with denser
regions attracting more matter, until the non-relativistic matter of the Uni-
verse was separated into clumps. This picture is actuall
y
how the
f
ormatio
n
of galaxies is expected to have taken place
(
Grupen et al.
,
2005
)
.Given
a
c
ertain amount of clumpiness observed today, it is possible to work out wha
t
d
ensity perturbations must have existed at the time o
f
last scatterin
g
base
d
on the Big Bang model of cosmology. Since these perturbations should corre
-
s
pond to re
g
ions o
f
di
ff
erent temperatures, the observed L
SS
o
f
the Universe
would impl
y
the existence o
f
temperature variations at a level o
f
about on
e
p
art in 1
0
5
o
f
T
γ
TT
i
n the microwave sky. Of course, this reasonable expectation
was verified by the COBE and WMAP measurements
.
Let us
g
ive a brie
f
summary o
f
the physics behind the
C
MB radiation
.
Before recombination, the baryons and photons were tightly coupled and
t
h
epertur
b
ations osci
ll
ate
d
in t
h
e potentia
l
we
ll
spro
d
uce
d
primari
ly by
t
h
e
d
ark matter perturbations
(
Nakamura
e
tal
.
, 2010
)
. After the decoupling of
photons from baryons, the baryons were free to collapse into those potentia
l
wells. The
C
MB is expected to carr
y
a record o
f
conditions at the time o
f
l
ast scatterin
g
, often called the primary anisotropies. It may also be affecte
d
by a time-varying gravitational potential
(
i.e., the integrated Sachs-Wolfe
effect or the so-called ISW rise
)
, gravitational lensing and scattering from a
h
omogeneous distribution of ionized gas at low redshifts. These effects ca
n
a
ll be computed by using the CMBFAST
(
Seljak and Zaldarriaga, 1996
)
or
CAMB
(
Lewis
et al.
,
2000
)
code based on the linear perturbation theory. The
a
ngular power spectrum of the CMB is usually plotted a
s
(
+1
)
C
C
/
(
2
π
)
versus t
h
emu
l
tipo
l
enum
b
e
r
,
as s
h
own in Fi
g
. 10.2, in w
h
ic
h
a
d
escriptio
n
o
f
the physics underlyin
g
the
C
MB anisotropies can be separated into thre
e
m
ain regions
(
Nakamur
a
e
tal
.
,
2010
):
•
t
he I
SW
rise with
1
0 and the Sachs-Wolfe plateau with 10
100
(
Sachs and Wolfe, 1967
)
at large angular scales
;
•
t
he acoustic
p
eaks with 10
0
1000 at sub-de
g
ree an
g
ular scales
;
•
th
e
d
amping tai
l
wit
h
1
000 at very tiny angular scales
(
Silk, 1968
)
.
In particular, the acoustic peaks represent the oscillations of the photon
-
baryon fluid around the decouplin
g
time of photons from matter.
A
smen-
tioned above, the density fluctuations in the early Universe gave rise to grav-
i
tational instabilities. When a kind of matter fell into these gravitational
potential wells, it was compressed and thereb
y
heated up. This hot matte
r
radiated photons causing the plasma of baryons to expand, and then it cooled
d
own an
dh
ence pro
d
uce
dl
ess ra
d
iation. Wit
hd
ecreasin
g
ra
d
iation pressure
the irre
g
ularities could reach a point at which
g
ravity a
g
ain took over and
i
nitiated another compression phase
(
Grupen et al.
,
2005
)
. A competition
1
0.1 The
C
osmic Microwave Background 35
7
Fi
g
. 10.2 A theoretical CMB anisotropy power spectrum computed from CMB-
FA
S
T based on the standard
Λ
C
DM model
(
Nakamura
et al.
, 2010. With permissio
n
from the Institute of Physics
)
b
etween t
h
e
g
ravitationa
l
accretion an
d
ra
d
iation pressure resu
l
te
d
in
l
on
g
i-
tudinal acoustic oscillations in the photon-baryon fluid.
A
fter the decoupling
o
f
matter
f
rom radiation, the pattern o
f
acoustic oscillations became
f
roze
n
i
nto the
C
MB.
S
othe
C
MB anisotropies are actuall
y
a consequence o
f
soun
d
waves in the primordial baryon fluid. The first and most prominent acous-
tic pea
k
appears at
Δ
θ
∼
1
◦
or
∼
200, as one can see
f
rom Fi
g
. 10.2. It
s
position provides a measurement o
f
the total ener
g
y density parameter
Ω
of
the
U
niverse. The fact tha
t
Ω is very close to unity implies that the early
U
niverse should essentiall
y
be
fl
at, as predicted b
y
the in
fl
ation mechanism
.
The hei
g
ht of the first acoustic peak and the hei
g
htsandpositionsofother
pea
k
sa
ll
ow us to
d
etermine some ot
h
er important cosmo
l
ogica
l
parameters
,
s
uch as the Hubble parameter, the bar
y
on densit
y
and the amount o
f
cold
d
ark matter
(
Perkins, 2009
)
.
10.1.4 Neutrino S
p
ecies and Masse
s
A
n accurate measurement of the CMB anisotropy power spectrum may hel
p
dete
rmin
eo
r
co
n
st
r
a
in
t
h
e effect
i
ve
n
u
m
be
r
of
n
eut
rin
ofa
mili
es a
n
d
n
eut
rin
o
m
asses. Around the time of recombination
(
i.e.
,
t
rec
∼
3
×
1
0
5
yr
o
r
T
rec
TT
∼
0
.
3
eV
)
, the total energy density of relativistic particles
(
photons and neutrinos
)
c
an be ex
p
ressed as
f
ollows:
ρ
r
+
ρ
ν
=
ρ
r
1+
7
8
4
11
4
/
3
N
C
MB
ν
N
N
,
(
10.15
)
358 10 Neutrinos and
C
osmological
S
tructures
( )
&*1
(
"Y!
%Π Μ
Ν
\
Ν
\
Ν
\
( )
&*1
(
"Y!
%Π Μ
Ν
2 ^
Ν
Fig. 10.3 How sensitive the power spectrum of CMB anisotropies is to the effective
number of relativistic neutrino species
(
left
)
and to the fraction of free or interacting
neutrinos
(
right
)
, where the crosses denote the WMAP data and other parameters
of the standar
d
ΛCDM model have been fixed (Strumia and Vissani, 2006)
whe
r
e
ρ
r
is the energy density of photons
,
N
CM
B
ν
NN
de
n
otes t
h
ee
ff
ect
i
ve
n
u
m
-
ber of neutrino species relevant to the CMB, and Eq.
(
10.1
)
has been use
d
by replacing “3” wit
h
N
C
M
B
ν
NN
. There are two different wa
y
s to reconstruc
t
t
h
eva
l
ue o
f
N
C
MB
ν
N
N
from the observational data on CMB anisotropies
(
Stru-
m
ia and Vissani, 2006
)
:
(
a
)
a determination of the energy density
ρ
r
+
ρ
ν
which should significantly contribute to the measurable expansion rate of the
U
niverse aroun
d
recom
b
ination,
b
ecause recom
b
ination too
k
p
l
ace s
l
i
gh
t
l
y
l
ater than a transition
f
rom the radiation-dominated Universe to the matter-
d
ominated one;
(
b
)
a determination of the energy density of freely-movin
g
n
eutrinos which could slow down the
g
rowth o
f
matter perturbations and
the formation of galaxies and other structures
(
see Section 10.2 for more dis
-
c
ussions
)
. Fig. 10.3 illustrates how the power spectrum of CMB anisotropies
c
hanges with the effective number of relativistic neutrino species
(
left panel
)
a
nd with the fraction of freely-moving or interacting neutrinos
(
right panel
),
w
h
ere ot
h
er cosmo
l
ogica
l
parameters in t
h
estan
d
ar
d
Λ
C
DM model have bee
n
fixed
(
Strumia and Vissani, 2006
)
3
.
A
lthou
g
h current data on the CMB an
d
i
ts anisotropies are not precise enough to pin down the value of
N
C
M
B
ν
NN
,it
m
ust not be far away from
N
ν
NN
=3
.
T
he WM
A
P Collaboration has recentl
y
reported
N
C
M
B
ν
NN
=4
.
34
+
0
.
86
−
0
.
88
at the
6
8% confidence level based on a careful analysis of the 7-year data
(
Komats
u
et al
.
,
2010
)
. A similar study including the SDSS data on the DR7 hal
o
p
ower s
p
ectrum has arrived a
t
N
C
M
B
ν
NN
=
4
.
7
8
+
1
.
86
−
1
.
7
9
a
tthe95
%
confidence level
(
Hamann et al., 2010a
)
. These preliminary results seem to favo
r
N
C
MB
ν
NN
>
3
,
3
N
ote that the standard Big Bang model of cosmology is often referred to as
t
h
e
Λ
C
DM model
,
where the abbreviation
Λ
C
DM means a combination o
f
the
cosmological constant
Λ
a
nd cold dark matter
.
10.2 Large-scale
S
tructures and Dark Matter 35
9
i
mp
l
yin
g
t
h
at one or more steri
l
e neutrinos wit
h
very sma
ll
masses mi
ght
exist and contribute to the total ener
g
y density of relativistic particles. I
f
t
h
is is t
h
e case
,
suc
h
steri
l
e neutrinos s
h
ou
ld
mix wit
h
t
h
ree active neutrinos
a
nd could be thermall
y
excited b
y
the interpla
y
o
f
oscillations and collision
s
(
Kainulainen, 1990
)
. Their mass scale is likely to lie in the sub-eV range an
d
their family number might be one or two
(
Haman
n
et al
., 2010b
)
. A mild
c
onc
l
usion is t
h
at t
h
e present cosmo
l
o
g
ica
l
constraints cannot
b
euse
d
as a
n
a
rgument against the existence of light sterile neutrinos.
T
he Planck spacecraft, which is taking data on the CMB with a muc
h
better de
g
ree of accuracy than the WM
A
P satellite, is expected to prob
e
N
C
MB
ν
N
N
up
to the
p
recision
Δ
N
C
MB
ν
NN
=
±
0
.
2
6. Therefore, the ongoing Planck
m
easurement has a
g
ood chance to con
fi
rm or rule out the hypothesis o
f
l
ight and cosmologically friendly sterile neutrinos
(
Haman
n
et al.
,
2010b
)
.I
f
the
C
ν
B consists of both active and sterile neutrinos, how to directly detec
t
t
h
em wi
ll b
ea
g
reat c
h
a
ll
en
g
e. It is in princip
l
e possi
bl
eto
d
irect
l
ypro
be
the active and sterile com
p
onents o
f
the
C
ν
B
via the ca
p
ture o
f
relic electro
n
n
eutrinos by means of radioactive beta-decaying nuclei
(
L
i
et a
l.
,
2010
)
,suc
h
as
ν
e
+
3
H
→
3
He +
e
−
(
signal
)
against
3
H
→
3
H
e+
e
−
+
ν
e
(
background
).
Note that a
p
recision measurement of the CMB and its anisotro
p
ies can
a
lso help determine or constrain the sum of light neutrino masse
s
m
i
(
for
i
=1
,
2
,
···
)
. When the temperature of the Universe was below
m
i
,t
h
e cor-
respondin
g
neutrino mass ei
g
enstate would be non-relativistic. Hence neutri
-
n
os should be counted in the budget of non-relativistic matter when
T
m
i
,
a
n
dbec
l
ass
i
fied as
r
ad
i
at
i
o
n
w
h
en
T
m
i
.S
uch a “leak”
f
rom one cate
g
or
y
to the other is a rather specific effect of finite neutrino masses
(
Lesgourgues
,
2
010
)
. By measuring the time of matter-radiation equality and other cosmo
-
l
o
g
ica
l
parameters, one may pro
b
et
h
is
l
ea
k
an
d
o
b
tain a
b
oun
d
on t
h
e tota
l
m
ass of non-relativistic neutrinos
>
m
i
=
9
4
h
2
Ω
ν
e
V
(
see Eq.
(
10.21
)
fo
r
m
ore discussions
)
. The latter has been well constrained by means of curren
t
WM
A
P data combined with the distance information. Given the standard
Λ
C
DM model with
k
= 0, an analysis of the 7-year WMAP data yields
>
m
i
<
1
.
3eVatthe95
%
confidence level
(
Komats
u
et al.
, 2010
)
.Afte
r
the latest distance measurements
f
rom the bar
y
on acoustic oscillations in
the distribution of galaxies
(
Percival
e
tal
.
,
2010
)
and the Hubble constant
m
easurement
(
Ries
s
et al.
, 2009
)
are also taken into account, one arrives a
t
>
m
i
<
0
.
58 eV at the same confidence level
(
Komats
u
et al., 2010
)
.This
i
s currently the best upper limit on the sum of neutrino masses withou
t
i
n
f
ormation on the
g
rowth o
f
structure.
10.2 Lar
g
e-scale Structures and Dark Matter
TheLSSintheUniverse
(
e.g., galaxies, galactic clusters, superclusters, voids
,
etc.
)
should be seeded by some primordial density perturbations arising from
q
uantum fluctuations during the period of inflation. The latter provides
a
360 10 Neutrinos and
C
osmological
S
tructures
n
atural mechanism to solve the horizon problem, the
fl
atness problem an
d
the ma
g
netic monopole problem in cosmolo
g
y. Massive neutrinos must hav
e
played a role in the formation of the LSS, and they can leave a small imprint
i
n the measurable amount o
f
the
g
ravitational clusterin
g
o
fg
alaxies. In thi
s
c
ase it is possible to obtain a robust cosmological bound on the absolute mass
s
cale of light active and sterile neutrinos.
10.2.1 Inflation and Density Fluctuation
s
The mechanism of inflation, which describes an exponentially acceleratin
g
phase in the very early Universe, was originally proposed by Alan Guth to
s
olve the magnetic monopole problem
(
Guth, 1981
)
. Magnetic monopoles
were suggested by Paul Dirac a long time ago
(
Dirac, 1931
)
,andtheirex
-
i
stence is definitely predicted in grand unified theories
(
GUTs
)
. A magneti
c
m
onopole should be stable and its mass should be around the
G
UT scal
e
Λ
G
UT
∼
1
0
16
GeV, but nobody has yet succeeded in observing it.
M
odel-independently, the elegant idea of inflation
(
Guth, 1981; Linde,
1
982; Albrecht and Steinhardt, 1982
)
was motivated to resolve the cosmi
c
h
orizon and flatness
p
roblems. The horizon
p
roblem arises on account o
f
the observed isotropy o
f
the
C
MB radiation out o
f
the lar
g
est an
g
les in the
m
icrowave sky. The horizon at the time o
f
the decouplin
g
between matter an
d
radiation or equivalently the formation of the CMB
(
i.e.
,
t
d
e
c
≈
3
.
8
×
1
0
5
yr
,
c
orrespon
d
in
g
t
o
T
dec
TT
≈
0
.
2
6
e
V
an
d
z
dec
≈
1
100
)
was of
O
(
10
2
)
Mpc in size,
s
ubtendin
g
an an
g
le of about
1
◦
on the Earth toda
y
. It is therefore difficult t
o
u
nderstand how the large-angle uniformity in temperature could have been
a
chieved via some causal processes. In other words, the unexplained uni
f
or
m
temperature in re
g
ions that appear to be causally disconnected is referre
d
to as the horizon problem
(
Grupe
n
et a
l.
,
2005; Perkins, 2009
)
. The flatnes
s
pro
bl
em is associate
d
wit
h
to
d
ay
’
so
b
servation t
h
at t
h
e tota
l
ener
g
y
d
ensity
of the Universe is very close to the critical energy density
(
i.e.
,
Ω
≡
ρ
/
ρ
c
≈
1)
.
With the help of Eq.
(
9.10
)
, one ha
s
Ω
−
1=
k
˙
R
2
∼
*
kt
2
/
3
(
matter-dominated era,
R
∼
t
2
/
3
)
,
kt
(
radiation-dominated era
,R
∼
t
1
/
2
)
.
(
10.16
)
H
enc
e
Ω
(
t
0
)
≈
1 at present
(
roughly matter-dominated
)
implies tha
t
Ω
(
t
)
m
ust have been very much closer to unity at very early times of the Univers
e
(
essentially radiation-dominated
)
, and the level of fine-tuning o
f
Ω
(
t
)
→
1at
the Planck time would be o
f
O
(
10
−
60
)(
Grupe
n
e
tal.
,
2005
)
. This flatness
pro
bl
em imp
l
ies t
h
at a Universe wit
h
non-vanis
h
ing curvatur
e
k
to
d
ay
h
as t
o
require ver
yfi
nel
y
-tuned initial conditions, which seem to be ver
y
unnatural.
Let us take a look at how inflation works to solve the aforementioned
p
rob
-
l
ems. By de
fi
nition, in
fl
ation means a short period o
f
acceleratin
g
expansio
n
i
n the very early Universe. I
f
this period was dominated by the vacuum-ener
g
y
t
er
m
ρ
v
i
n the Friedmann equation as given in Eq.
(
9.8
)
, then the Univers
e
10.2 Large-scale
S
tructures and Dark Matter 36
1
wou
ld
un
d
er
g
o an exponentia
l
expansion
d
escri
b
e
db
y
H
=
˙
R
/R
=
Λ/
3
wit
h
Λ
=
8π
G
N
ρ
v
b
ein
g
the cosmolo
g
ical constant. Namely,
R
(
t
)=
R
(
t
i
)
exp
[
H
(
t
−
t
i
)]
,
(
10.17
)
where
t
i
denotes the initial time of inflation. In a s
p
ecific inflation model it i
s
possi
bl
eto
d
irect
l
y
l
in
k
ρ
v
t
ot
h
e potentia
l
V
(
φ
)
of a scalar field
φ
(
x
)(
Guth
,
1
981; Linde, 1982; Albrecht and Steinhardt, 1982
)
. The form of
V
(
φ
)
can be
taken in such a way that the inflaton field first settles down into a metastable
state o
r
fa
l
se vacuu
mn
ea
r
φ
≈
0
and then e
ff
ectively “rolls” down to the tru
e
vacuum
(
Grupen
et al.
,
2005
)
. A combination of Eqs.
(
10.16
)
and
(
10.17
)
yie
ld
s Ω
(
t
)
−
1
∝
e
xp
[
−
2
H
(
t
−
t
i
)]
,wher
e
H
i
saconstantan
d
t
runs from th
e
i
ni
t
i
al t
im
e
t
i
to t
h
efi
n
a
l
t
im
e
t
f
of
in
flat
i
o
n. Thi
s
r
esu
l
tte
ll
sust
h
at
Ω
(
t
)
wa
s
exponentially driven towards unity durin
g
inflation. So one may easily obtain
Ω
(
t
f
)
≈
1
only i
f
H
(
t
f
−
t
i
)
o
r
ρ
v
i
s sufficiently large. The flatness proble
m
c
an there
f
ore be resolved in this wa
y
,sinc
e
Ω
(
t
0
)
≈
1
o
b
serve
d
to
d
a
y
must
be intrinsicall
y
associated wit
h
Ω
(
t
f
)
≈
1
as a conse
q
uence of the evolutio
n
of the
U
niverse fro
m
t
f
to
t
0
t
h
roug
h
t
h
era
d
iation- an
d
matter-
d
ominate
d
epochs. Note that a re
g
ion o
f
size
D
(
t
)
in the physical coordinate syste
m
would ex
p
and in
p
ro
p
ortion to the scale facto
r
R
(
t
)
during inflation. Henc
e
the s
iz
e
D
(
t
i
)
of a region, which was in causal contact at the beginning o
f
i
n
fl
ation, would becom
e
D
(
t
f
)=
D
(
t
i
)
exp
[
H
(
t
f
−
t
i
)]
at the end of inflation.
The
p
resent size D
(
t
0
)
of this region, in which everywhere has been causally
c
onnecte
d
,must
b
eintrinsica
ll
yre
l
ate
d
t
o
D
(
t
f
)
and thus can be sufficiently
l
arge
(
e.g., much larger than the current Hubble distance
ct
0
≈
c/H
0
HH
∼
10
2
6
m)
by adjusting the exponential factor exp
[
H
(
t
f
−
t
i
)]
. So the currentl
y
visible Universe, includin
g
the entire sur
f
ace o
f
last scatterin
g
, can easily
fit
i
nto a much lar
g
er re
g
ion which has been in causal contact and has bee
n
a
t the same temperature
(
Grupe
n
et a
l.
, 2005
)
. This is just the solution t
o
the horizon problem. In other words, in
fl
ation provides a natural mechanism
to ensure the isotrop
y
of the CMB radiation in the entire microwave sk
y
to
a
high degree of accuracy. A solution to the magnetic monopole problem is
a
lso strai
g
ht
f
orward.
O
ne may simply assume that ma
g
netic monopoles wer
e
produced before or durin
g
theperiodofinflation.InaGUTframeworkwhose
typica
l
temperature an
d
time sca
l
es wer
e
T
∼
10
16
G
e
V
an
d
t
∼
10
−
39
s,
f
or example, one expects that in
fl
ation should be takin
g
place durin
g
the
p
eriod from
t
i
∼
1
0
−
38
s
t
o
t
f
∼
1
0
−
36
s. The number densities of magneti
c
m
onopoles at the beginning and end of inflation are related to each other vi
a
n
(
t
f
)=
n
(
t
i
)
exp
[
−
3
H
(
t
f
−
t
i
)]
, just because the volume containing a given
n
umber of magnetic monopoles increase in proportion t
o
R
3
(
t
)
.Henc
e
n
(
t
f
)
m
ust be vanishin
g
ly small i
f
H
(
t
f
−
t
i
)
is sufficiently large as required to solv
e
the
fl
atness and horizon problems.
S
ince today’s number density o
f
ma
g
netic
m
ono
p
ole
s
n
(
t
0
)
must be intrinsically associated with
n
(
t
f
)
, we expect that
the in
fl
ationary expansion o
f
the Universe mad
e
n
(
t
0
)
so small that there
m
i
g
htnotevenexistasin
g
le ma
g
netic monopole in the observable Universe
.
T
h
at is w
h
yone
h
as never seen a magnetic monopo
l
e
.
362 10 Neutrinos and
C
osmological
S
tructures
A
fter inflation the ener
gy
ρ
v
≈
V
(
0
)
was transferred to particles such as
photons, electrons and neutrinos, but their ener
g
y density turned to decrease
a
s the Universe continued to expand in a relatively mild way. Because infla
-
tion itsel
f
mi
g
ht not simultaneously end everywhere
4
, the commencement o
f
this energy density decrease was somewhat delayed in regions where infla
-
tion went on a bit longer. As a result, the variation in the time of the end
o
f
in
fl
ation provides a natural mechanism to interpret spatial variations in
the energy density. Such primordial density fluctuations were later on ampli-
fied by gravity and finally led to the LSS observed today, such as galaxies,
g
alactic clusters and superclusters.
A
s discussed in Section 10.1.3, quantu
m
fluctuations at the very beginning of inflation were also responsible for the
observed
C
MB anisotropies at the
O
(
1
0
−
5
)
level in the microwave sky. The
y
would become
f
rozen densit
yfl
uctuations when the
y
were in
fl
ated be
y
ond th
e
c
ausal horizon
(
Perkins, 2009
).
T
he existence o
f
the L
SS
is naturally attributed to the primordial den-
s
it
yfl
uctuations. To be more explicit, let us consider the relative di
ff
erenc
e
b
etween t
h
e
d
ensity at a given position an
d
t
h
e average
d
ensity
:
δ
(
x
)
≡
ρ
(
x
)
−
ρ
ρ
.
(
10.18
)
N
ow that this densit
y
contrast inhabits a Universe which is isotropic and
h
omogeneous on sufficiently large scales, its statistical nature should also b
e
i
sotropic and homo
g
eneous.
O
ne usually describes
δ
(
x
)
as a Fourier series
with periodic boundar
y
conditions
:
δ
(
x
)
=
δ
k
e
i
k
·
x
,
(
10.19
)
w
h
e
r
et
h
esu
mi
sta
k
e
n
ove
r
a
ll
va
l
ues of
k
=
(
k
x
,k
y
,
k
z
)
which fit into a cub
e
of s
iz
e
L
a
n
d volu
m
e
V
=
L
3
(
e.g.,
k
x
=
2
π
n
x
/L
w
i
th
n
x
=0
,
±
1
,
±
2
,
···
;
a
nd similarl
y
fo
r
k
y
an
d
k
z
)
. Averaging over all directions yields the average
m
a
g
nitude o
f
the Fourier coe
ffi
cients as a
f
unction o
f
k
=
|
k
|
(
Grupen
et al
.
,
2
005
)
. So the matter power spectrum is defined a
s
P
(
k
)
≡|
δ
k
|
2
=
|
δ
k
|
2
,
(
10.20
)
which measures the level o
f
structure at a wavelen
g
th
λ
=
2
π
/
k
.
In m
ost
of the structure formation models one arrives at a
p
rimordial
p
ower law
of t
h
efo
rm
P
(
k
)
∼
k
n
,w
h
er
e
n
=
1
d
enotes t
h
e sca
l
ar spectra
l
in
d
ex an
d
y
ields a scale-invariant Harrison-Zeldovich spectrum. Most o
f
the in
fl
ationar
y
m
o
d
e
l
spre
d
ic
t
|
n
−
1
|
3
%,
and the exact value of
n
d
epends on the form
o
f
the potentia
l
V
(
φ
)(
Peacock, 1999; Liddle and Lyth, 2000
)
. For example,
a
nanal
y
sis of the 5-
y
ear WM
A
P data
y
ield
s
n
=
0
.
96
3
+0
.
0
1
4
−
0
.
0
1
5
b
ased on the
s
tan
d
ar
d
Λ
C
DM model
(
Nakamur
a
e
ta
l.
,
2010
).
4
D
ue to
q
uantum
fl
uctuations, the value o
f
the in
fl
aton
fi
el
d
φ
at the onset o
f
inflation would not be exactly the same at all places (Grupen et a
l.
, 2005).
10.2 Large-scale
S
tructures and Dark Matter 36
3
I
n short, the main success o
f
in
fl
ationar
y
models is that the
y
provide
u
s with a simple but d
y
namical interpretation of specific initial conditions o
f
the Universe such that the horizon, flatness and magnetic monopole problems
c
an naturall
y
be resolved. Moreover, in
fl
ation o
ff
ers a natural mechanism to
explain the primordial density fluctuations which finally grew into the LS
S
s
uc
h
as ga
l
axies an
d
c
l
usters o
b
serve
d
to
d
ay. We s
h
a
ll
procee
d
to
d
iscuss t
he
i
ndispensable role o
f
dark matter in the
f
ormation o
f
the L
SS.
1
0
.2.2 L
SS
and Dark Matte
r
The LSS observed today should most likely have evolved from gravitationa
l
i
nstabilities which can be traced back to the primordial densit
yfl
uctuation
s
i
n the very early Universe. Such small perturbations in the ener
g
y density
were amplified by gravity, and in the course of time they would collect mor
e
a
nd more matter and lead to the
f
ormation o
f
the L
SS
. The latter require
s
a
sufficient amount of mass, otherwise the ori
g
inal density fluctuations coul
d
n
ever have been transformed into distinct mass aggregations
(
Grupe
n
et al
.
,
2
005
)
. However, the amount of visible matter itself has been found to b
e
i
nsufficient for the Universe to reach the critical energy density today
(
i.e.,
Ω
≈
1
)
. Non-baryonic dark matter is therefore needed in order to understand
the d
y
namics o
f
the Universe and its evolution.
I
n 1933, Fritz Zwicky found that luminous galaxies in the Coma cluster
o
fg
alaxies moved
f
aster than one would have expected i
f
they had only
f
elt
the gravitational attraction from other visible objects
(
Zwicky, 1933, 1937
)
.
H
e was the first to infer the existence of dark matter by means of the viria
l
t
h
eorem. In 1970 an
d
1980, Vera Ru
b
in an
dh
er co
ll
a
b
orators measure
d
t
h
e
rotation curves o
f
individual
g
alaxies and
f
ound
f
urther evidence
f
or invisibl
e
m
atter
(
Rubin and Ford, 1970; Rubi
n
e
tal.
,
1980
)
. Recent observations of
c
lusters o
fg
alaxies yiel
d
Ω
DM
≈
0
.
2ascompare
d
wit
h
Ω
=
1
f
or a
fl
at
U
ni
-
verse. They include measurements of the peculiar velocities of
g
alaxies in th
e
c
luster; measurements of the X-ray temperature of hot gas in the cluster; an
d
m
ost directly, studies of
(
weak
)
gravitational lensing of background galaxie
s
on the cluster
(
Nakamura
e
tal
.
,
2010
)
. A particularly compelling example
i
s the bullet cluster
(
1E0657-558
)
which passed through another cluster re
-
c
ently. The hot
g
as
f
ormin
g
most o
f
the baryonic mass o
f
these two cluster
s
was shocked and decelerated, whereas the galaxies in them proceeded on bal
-
l
istic trajectories. Gravitational lensing indicates that most of the total mass
o
f
the clusters also moved ballistically, implyin
g
that the sel
f
-interaction o
f
d
ark matter should be quite weak
(
Clow
e
et al.
,
2006
;
Nakamur
a
et al., 2010
)
.
In a
dd
ition, current cosmo
l
o
g
ica
l
o
b
servations
h
ave c
l
ear
l
y
d
emonstrate
d
t
he
existence o
f
alar
g
equantityo
f
cold dark matter and accurately determine
d
the value of Ω
C
DM
. For example, an analysis of the 5-year WMAP data
yie
ld
s
Ω
C
D
M
h
2
=
0
.
110
±
0
.
006
wit
h
h
=0
.
72
±
0
.
0
3
(
Dunkle
y
et al
.,
2009
;
Ko
m
atsu
et al.
,
2009
)
. This result can be compared with the total matte
r
d
ensity Ω
m
a
nd the baryonic matter densit
y
Ω
B
obtained from the same
364 10 Neutrinos and
C
osmological
S
tructures
a
na
ly
sis
b
ase
d
on t
h
estan
d
ar
d
Λ
C
DM model, as shown in Table 10.1. We
see
Ω
C
D
M
/Ω
m
≈
83
%
an
d
Ω
B
/Ω
m
≈
1
7%
;
i.e.
,
cold dark matter dominates
today’s matter content of the Universe. So somebody said, “If it’s not
d
ar
k
,
it does
n
’t
matter
” (Grupen
rr
et al
., 2005
)
. Note that the average density of
c
old dark matter in the “neighborhood” of our solar system is roughly equa
l
to that of luminous matter
(
stars, gas and dust
)
:
ρ
l
oca
l
C
D
M
≈
0
.
3G
e
V
c
m
−
3
(
Kamionkowski and Kinkhabwala, 1998
)
and thus much larger than today’
s
c
ritical energy density of the Universe ρ
c
≈
5
.
4
×
10
−
6
G
eV c
m
−
3
.
Table 10.1 Some important cosmological parameters determined from an analysis
of the 5-year WMAP data based on the standard
Λ
C
DM model
(
Nakamur
a
et al.
,
2010. With permission from the Institute of Physics
)
. The cold dark matter density
Ω
C
D
M
i
sgivenby
Ω
CDM
=
Ω
m
−
Ω
B
−
Ω
ν
P
arameter
V
a
l
ue
The Hubble
p
arameter h 0
.
72
±
0
.
03
The total matter densit
y
Ω
m
Ω
m
h
2
=0
.
133
±
0
.
006
T
h
e
b
aryon
d
ensity Ω
B
Ω
B
h
2
=
0
.
0
22
7
±
0
.
0006
T
h
e vacuum ener
g
y
d
ensit
y
Ω
v
Ω
v
=
0
.
74
±
0
.
03
T
h
era
d
iation
d
ensit
y
Ω
r
Ω
r
h
2
=
2
.
47
×
10
−
5
The neutrino densit
y
Ω
ν
Ω
ν
h
2
=
>
m
i
/
(
94 eV
)
T
he nature of dark matter remains a m
y
ster
y
in particle ph
y
sics and cos
-
m
ology. A candidate for non-baryonic dark matter must satisfy the followin
g
three conditions
(
Nakamura
et al
.
,
2010
)
:
(
a
)
it must be stable on cosmologi
-
c
al time scales, otherwise it would have decayed by now;
(
b
)
it must interac
t
very wea
kl
ywit
h
e
l
ectromagnetic ra
d
iation, ot
h
erwise it wou
ld
not
b
equa
l-
i
fied as dark matter; and
(
c
)
it must have the right relic density. Possible
c
andidates of cold dark matter include
p
rimordial black holes, axions and
weakly interacting massive particles
(
WIMPs
).
• Primordial black holes were proposed in a few cosmological models as
a
candidate for dark matter
(
Kohr
i
et al.
,
2008
)
. They must have formed be
-
fore the BBN epoch, otherwise the
y
would have been counted as bar
y
oni
c
m
atte
rin
stead o
f
co
l
dda
rk m
atte
r.
•
A
xions were first postulated to resolve the stron
g
CP problem in quantu
m
chromodynamics
(
Peccei and Quinn, 1977
)
. This kind of pseudo-Nambu-
G
oldstone bosons may result from the spontaneously broken Peccei-Quin
n
U
(
1
)
symmetry at very high energies. Axions could constitute cold dar
k
matter if they were non-thermally produced in the early Universe and
t
heir masses lie in the sub-eV range
(
e.g., from
O
(
10
−
6
)
eV to
O
(
1
)
eV
)
.
•
W
IMPs are b
y
de
fi
nition a kind o
f
particles whose interactions with ordi-
nary matter are weak and whose masses lie in the range between 10 GeV
10.2 Large-scale
S
tructures and Dark Matter 36
5
and a
f
ew TeV. WIMPs could constitute cold dark matter i
f
the
y
wer
e
t
hermall
y
produced in the earl
y
Universe and then froze out after the
t
emperature was
b
e
l
ow t
h
eir masses. T
h
e
l
ig
h
test supersymmetric parti-
cle
(
e.g., neutralino
)
is currently the best motivated and widely studied
W
IMP candidate
(
Nakamur
a
et al.
,
2010
)
.
These candidates
f
or cold dark matter are all detectable by means o
f
curren
t
experimental techniques, but no convincin
g
evidence
f
or any o
f
them has s
o
f
ar been found. The present dark matter searches include direct searches
,
i
ndirect searches and collider searches. Takin
g
WIMPs
f
or example, one ma
y
d
irectly search for them by detectin
g
their interactions with ordinary matter
through elastic scattering on nuclei; or indirectly search for them by detect
-
i
n
g
t
h
eir anni
h
i
l
ation pro
d
ucts suc
h
as neutrinos,
g
amma rays, positrons,
a
ntiprotons and antinuclei; or look for their si
g
natures at hi
g
h-ener
g
y collid-
ers such as the Large Hadron Collider
(
LHC
)
. A combination of all of the
m
i
s necessary in order to pin down the nature o
f
WIMPs on solid
g
round
.
I
n general, candidates for dark matter are subdivided into three cate
-
gories: “
h
ot
”
,“co
ld”
an
d
“warm
”
partic
l
es. Primor
d
ia
lbl
ac
kh
o
l
es, axion
s
a
nd WIMPs belon
g
to cold dark matter. The typical example o
f
hot dark mat-
ter is light neutrinos. Sterile neutrinos with masses of
O
(
1
)
keV are possible
to co
n
st
i
tute wa
rm
da
rk m
atte
r. A l
ot o
fm
o
r
eo
rl
ess
“
e
x
ot
i
c
”
ca
n
d
i
dates
f
or
d
ark matter have also been proposed in the literature.
A
lthou
g
h the nature
of cold dark matter remains unclear
,
we believe that it dominates the matte
r
c
ontent o
f
the Universe and is primarily responsible
f
or the
f
ormation o
f
the
observed L
SS
. The status and
p
ros
p
ects o
f
direct and indirect dark matter
s
earches have recently been reviewed by the Particle Data Group
(
Nakamura
et al
.
,
2010
)
and other authors
(
Feng, 2010
)
.
1
0
.2.
3C
onstraints on Neutrino Masses
Given the Cν
B
tem
p
erature
T
ν
TT
≈
1
.
9
45 K today, the average three-momentum
of each relic neutrino is expected to b
e
p
ν
=3
T
ν
TT
≈
5
×
10
−
4
eV
as given
i
nEq.
(
9.54
)
. Hence neutrinos have become non-relativistic today if their
m
asses are larger tha
n
p
ν
. In this case the
p
resent
C
ν
B
contribution to th
e
total ener
g
y density o
f
the Universe i
s
5
5
I
f one of the neutrinos has a mass much smaller tha
n
p
ν
,
it ma
y
remai
n
relativistic today. But its contribution to the sum of all neutrino masses is insignif-
icant, so Eq.
(
10.21
)
remains valid as a good approximation. On the other hand,
light sterile neutrinos and antineutrinos should be most likely to stay in full thermal
equilibrium in the early Universe provided their mixing with active neutrinos and
antineutrinos is not strongly suppressed
(
Hannestad and Raffelt, 1999
)
. In this case
their number densities are expected to be equal to those of active neutrinos and
antineutrinos, because the calculation of
n
ν
i
a
n
d
n
ν
i
h
as not
h
ing to
d
owit
h
t
he
flavor properties. Therefore, the sum in Eq.
(
10.21
)
is over both active and sterile
neutrinos only if they are sufficiently light
.