Astroparticle Course 1
Standard CosmologyStandard Cosmology
The fate of the universeThe fate of the universe
Red shiftRed shift
Hubble lawHubble law
Age of the universeAge of the universe
Problems of standard cosmologyProblems of standard cosmology
Astroparticle Course 2
LongLong--term behaviour of a(t) (I)term behaviour of a(t) (I)
The full solutions to the Friedmann equations can be quite
complicated. However, it is possible to study a few generic case, in
particular starting from the present time (⇒ radiation plays no role)
and considering t→∞. In case Λ=0,
3
3
0
0a
am ρρρ == k
a
aGa N −=
3
8 0302 ρπ
&
0=k3/2
)( tta ≈ Einstein-de Sitter universe
1−=k ttat
≈∞→
)(lim
since at the end the matter term becomes
negligible
1+=k3
8 030
max
ρπ aGa N=
imposing that the right member be 0
Astroparticle Course 3
LongLong--term behaviour of a(t) (II)term behaviour of a(t) (II)
In case Λ≠0,33
82
0302 a
ka
aGa N Λ
+−=ρπ
&
0<Λ
)(3/
00)()(
ttetata
−Λ≅
Λ contribution is negligible for small a(t) and dominates for large a(t). This is puzzling: why is it that we live at a so special epoch,
when ρΛ is of the order of ρm?
033
82max
max
030 =
Λ+−
ak
a
aGN ρπ
oscillating universe
1,00 −=>Λ k
exponential expansion
10 =>Λ kthere exists Λc giving a static solution,
Λ>Λc ⇒ expansion forever, Λ<Λc ⇒ forbidden range for a(t)
Problem of
coincidence
Astroparticle Course 4
Solutions of the Friedmann equationSolutions of the Friedmann equation
Λ=0, k=-1
Λ=0, k=0
Λ=0, k=+1
Λ>0, k=0
tH=H0-1
Astroparticle Course 5
Redshift (I)Redshift (I)
Let us consider the geodesic motion of a particle that is not
necessarily massless,
0=Γ+λλ
ανµ
να
µ
d
dxu
d
du
where uµ=dxµ/ds and λ is some affine parameter. If λ is the proper length, s, the geodesic equation becomes
01
0=+ u
a
a
ds
ud
u
r&r
Astroparticle Course 6
Redshift (II)Redshift (II)
Since u0=dt/ds, we get
that is the three-momentum of a freely-propagating particle in an expanding universe red shifts. This occurs for photons too,
a
a
u
u &r
r
−=
•
1−∝ aur 1−∝= aump
rr
zta
ta
t
t+== 1
)(
)(
)(
)(
1
0
1
0
λ
λ λ(t0)=received wave length λ(t1)=emitted wave length
z=redshift
The redshift z is often used in place of the scale factor.
Astroparticle Course 7
Redshift (III)Redshift (III)
Consider a fixed comoving volume of the universe, i.e. a volume
specified by fixed values of the coordinates, from which one
may obtain the physical volume at a given time t by multiplying
by a(t)3. Given a fixed number of dust particles (of mass m) within this comoving volume, the energy density will then scale
just as the physical volume, i.e. as a(t)−3, in agreement with the
equation of state with w = 0.
In the case of radiation, because of the redshift, the energy
density of a fixed number of photons in a fixed comoving
volume drops with the physical volume (as for dust) and by an
extra factor of the scale factor, as the expansion of the universe stretches the wavelengths of light. Thus, the energy density of
radiation will scale as a(t)−4, once again in agreement with the
equation of state with w = 1/3.
Astroparticle Course 8
Linear HubbleLinear Hubble’’s laws law
In a uniformly expanding
universe, in first approxi-
mation, Hubble’s law is valid,
Hdv =
Let us consider a bidimensional analogy. The distance between
two points on a 2-sphere at t1[t2] is d1=a(t1)s[d2=a(t2)s], where s is the arclength distance between the points. Then
Hdsastt
tata
tt
ddv
tttt==
−
−=
−
−=
→→&
12
12
12
12 )()(limlim
1212
Astroparticle Course 9
Deviations from linear HubbleDeviations from linear Hubble’’s laws law
Actually, acceleration equation
implies that, if matter drives the
expansion, the rate is presently
decelerating.
+−−−+= ...)(
2
1)(1)()(
2
0
2
00000 ttHqttHtata
)3(3
4p
G
a
a N +−= ρπ&&
However, observations indicate that vacuum energy is
dominating and the expansion is accelerating. In any case, we
may expand the scale factor in a Taylor series around the present time, t0,
which defines the deceleration parameter, q0. 2
00
0
0Ha
aq
&&−=
ρ>0 p∼0
a0=a(t0)
Astroparticle Course 10
Deceleration parameterDeceleration parameter
From the acceleration equation, the expression of ρc, and the equation of state for each different contribution (matter, radiation, and vacuum) to the universe energy density, we get
∑∑ +Ω=+=i
ii
i
iiN p
H
Gq )31(
2
1)3(
3
8
2
10002
0
0ωρ
π
Since a cosmological constant has ω=-1, q0 can be negative if
the energy density of the universe is dominated by ρΛ. In
general, acceleration of the expansion is possible for ωi=-1/3.
Astroparticle Course 11
RedRed--shift expansionshift expansion
Using the definition of red-shift, the expansion of a(t) gives
1
2
0
2
00000 ...)(
2
1)(1
)(1
−
+−−−+==+ ttHqttH
ta
az
...)(2
1)(112
0
2
00
00 +−
++−−=+ ttH
qttHz
...)()(2
1)(11
2
0
2
0
2
0
2
0000 +−+−+−−=+ ttHttHqttHz
...)(2
1)(2
0
2
0
0
00 +−
++−−= ttH
qttHz
Astroparticle Course 12
Particle horizonParticle horizonA light ray that moves radially (θ,φ=const) toward us, from r=re at time
te to r=0 at time t, along a geodesic
(ds2=0 ⇒ dt=a dr/(1-kr2)1/2), will satisfy∫∫ =
−
t
t
r
e
e
ta
dt
kr
dr
)'(
'
10 2
and the physical distance will be
∫∫ =−
=t
t
r
e
e
ta
dtta
kr
drtatd
)'(
')(
1)()(
0 2
Particle horizon is the farthest physical
distance we can observe in the past. It
is given by the limit te → 0, unless the integral does not converge (this does
∫=t
pta
dttatd
0 )'(
')()(
not happen in conventional cosmological theories). In a RD era,
dp=2t, while in a MD era dp≅3t.
Astroparticle Course 13
Event horizonEvent horizon
∫∫∞
=− t
r
ta
dt
kr
dro
)'(
'
10 2 ∫∞
=t
eta
dttatd
)'(
')()(
While particle horizon is the maximum distance wherefrom
signals can reach the observer, event horizon is the maximum
distance that signals we send can reach. It is determined by
In a matter dominated era, the integral diverges, so that there is no event horizon. But, in presence of a cosmological constant,
a(t) will eventually grow exponentially, giving a finite event
horizon. This means that, as time passes, all sources of light
outside our Local Group will move beyond de and become
unosservable.
Astroparticle Course 14
HorizonsHorizons
Let us remember the definition of particle and event horizon:
∫=t
pta
dttatd
0 )'(
')()( ∫
∞
=t
eta
dttatd
)'(
')()(
If a(t)∼tα, then∞=
−= )(
1)( td
ttd ep
α1<α
1)()(
−=∞=
α
ttdtd ep1>α
If a(t)∼eHt, thenH
tdH
etd e
Ht
p
1)(
1)( =
−=
Astroparticle Course 15
Distance expansionDistance expansion
Expanding left and right member of previous equation gives
...2
11
12
2++=
−
kr
kr...
61
3
0 2++=
−∫ ee
r
rk
rkr
dre
+−
++−−== ...)(
21)(1
1
)(
1
)(
1 2
0
2
0
0
00
0
0
0
ttHq
ttHata
a
ata
+−+−=∫ ...)(
2)
1
)(
20
0
eoeo
t
ttt
Htt
ata
dto
e
+−+−= ...)(
2)
1 20
0
eoeoe ttH
tta
rand finally
Astroparticle Course 16
Luminosity distance (I)Luminosity distance (I)
To obtain information on the cosmological parameters we use
measurements like the intensity of a source of known intrinsic
strangth, a standard candle. Given a source of absolute
luminosity L, its apparent luminosity at a distance d is l=L/(4πd2). At large distance this expression needs modification. Suppose
that Nγ photons was emitted isotropically at te. If there were no
expansion, a telescope of area A, at distance a(te)r, would see a
fraction of photons,
( )2
det
)(4 rta
A
N
N
eπγ
=
Due to the expansion, at detection time, to,( )2
det
)(4 rta
A
N
N
oπγ
=
Astroparticle Course 17
Luminosity distance (II)Luminosity distance (II)
Two additional effects have to be taken into account. First, all
photons will have energy red-shifted by the factor 1+z. Second,
the rate of arrival of individual photons is lower than the rate at
which they are emitted, since the time interval will be increased by the same factor, 1+z. This means that the flux measured at the
telescope will be
( ) 2224)1()(4 Lo
d
L
zrta
L
ππ=
+=F
which defines the luminosity distance, dL,
)1()( zrtad oL +=
Astroparticle Course 18
HubbleHubble’’s laws law
After some arithmetics,
...)(2
1)(22
0
0
0 +−
++−= eoeo ttH
qttHz
+
+−=− ...
21
1 20
0
zq
zH
tt eo
+−+−= ...)(
2
1 20
0
eoeoe ttH
tta
r
++−= ...)1(
2
11 2
0
00
zqzHa
re
+−+= ...)1(
2
11 2
0
0
zqzH
dL
deviation from the linear Hubble law
z∼v
Astroparticle Course 19
Standard candlesStandard candles
Establishing the distances to
other galaxies has been one
of the main efforts of
observational cosmology. For minimizing the
uncertainties introduced by
peculiar velocities, one needs to reach the Hubble flow, that is the
distance range where the velocities of the galaxies are dominated
by the cosmological expansion. To do that, one of the most
common methods exploits the homogeneity of the peak brightness of Type Ia supernovae (SNIa), which has about a 20%
intrinsic dispersion. Connecting the distance of the galaxy hosting
the supernova with the recession velocity (given by the red-shift),
one builds a Hubble diagram.
SNIa: a small star (white dwarf) which, accreting matter from a companion, overcomes the Chandrasekar limit
Astroparticle Course 20
Hubble diagramHubble diagram
Hubble diagram for high red-shift SNIa from the Supernova
Cosmology Project and low red-shift SNIa from the CfA and
Calan/Tololo Supernova Survey.
apparent magnitude
The data favour a
universe with a positive cosmological constant.
The analysis of SNIa
gives:
15.065.0
)05.0(02.065.0
0 ±−=
±±=
q
h
( )ΛΩ−Ω= 22
10 Mq
Astroparticle Course 21
Degeneracy in the SNIa analysisDegeneracy in the SNIa analysis
The SNIa data are
insufficient to break the degeneracy of the density
terms. The results can be
approximated by the linear
combination
1.02.06.08.0 ±−=Ω−Ω ΛM
Such a degeneracy can be
break only with the inclusion
of other kind of observation.
Astroparticle Course 22
Hubble parameter evolutionHubble parameter evolution
Remembering the rate of change of ρM and ρΛ with a(t), the definition of red-shift and curvature parameter,
( )( )03000
3
001 ΛΛΛ Ω++Ω=+
=+= z
a
aMcMM ρρρρρρ
( ) ( )( )200302
0
211 zzHH kM +Ω+Ω++Ω= Λ
( )202
0
2
0
2
0
21 zH
a
a
a
k
a
kk +Ω=
−=−
Astroparticle Course 23
Age of universeAge of universe
From the definition of the Hubble parameter and red-shift,
( )dt
dz
zz
dt
d
a
a
dt
dH
+−=+−=
=
1
1)1ln(ln
0
( ) ( )( ) 2/120030
0
1
11
)1(1
1
1
zzH
z
Hzdz
dt
kM +Ω+Ω++Ω
+−=
+−=
Λ
−
( ) ( )( )∫+Ω+Ω++Ω+
=
Λ
∞→
z
kM
zzzz
dz
Ht
0 2/1200300
0
'1'1)'1(
'lim
1
and the present age of the universe is
Astroparticle Course 24
Age of a flat universeAge of a flat universe
001 =Ω=Ω=Ω Λ kM ( ) 3
2
10 2/500 =+
= ∫∞
z
dztH
For H0=65 km s-1 Mpc-1, this gives an age of about 10 billion years.
The age of the universe for a
given ΩM is always larger in the flat case (with a
cosmological constant). Also,
the age is larger for smaller
values of ΩM.
Einstein-de Sitter universe
Astroparticle Course 25
The flatness problemThe flatness problem
From the Friedmann equation,
2
2
3
8
a
kGH N −= ρ
π2
2
3
81
8
3
aG
k
G
H
NN ρπρπ
−=
cρ
2
2
1
8
31
−− ≈−=−Ω α
ρπa
aG
k
N
α−a
This shows that at early times (small a) Ω must have been very close to unity. For example, at the time of nucleosynthesis:
Ω(t=1s)=1±10-16 and at the Planck time Ω(t=10-43s)=1±10-60.
60
2
0
2
0
1101
−− ≈
=
≈−Ω
PlT T
T
a
a
Pl
How is it possible a How is it possible a similar finesimilar fine--tuning?tuning?
Astroparticle Course 26
The horizon problemThe horizon problem
Let us consider two photons
which arrive to us today from
two different directions in the
sky. It is possible to consider
directions corresponding to regions causally decoupled
at the time of decoupling.
How is it possible How is it possible that the two that the two photons have the photons have the same T?same T?
Astroparticle Course 27
The The ΛΛ problemproblem
How is it possible to have a so How is it possible to have a so small dimensionless number?small dimensionless number?
Going back to the complete Friedmann equation, we see that the
comparison between the matter term and the cosmological
constant term is given by their ratio, κ,
323
82 Λ+−=
a
kGH N ρ
π
ρπκ
NG8
Λ=
Considering the value that κ has today, one obtains that at the Planck time it should have been 10-122!!!!
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