Download Cosmological solutions of the Einstein

Cosmological solutions of the Einstein-Friedmann equations
Summary of Friedmann’s equations
Ingredients are the Einstein equations of GRT:
Rµν −
1
R gµν − Λ gµν = κ T µν ,
2
with the cosmic ideal fluid energy-momentum tensor
T µν = (ρ + p) (t) uµuν − p(t) gµν .
The cosmological constant term Λ gµν nowadays is usually included as part of T µν
as an additional contribution to the energy density
ρ(t) → ρ(t) + ρΛ ; ρΛ =
Λ
,
κ
c 2009, F. Jegerlehner ≪x Lect. 7 x≫
Cosmology
claimed to represent the vacuum energy density. The ideal fluid tensor structure
then requires the vacuum energy, now usually called dark energy to satisfy the
exotic equation of state
pΛ = −ρΛ .
In the following we discuss the resulting set of Friedmann equations
Ṡ 2
c2
d
dS
= 3κ ρ S 2 − k
Friedmann equation
3
ρ S = −3 p S 2 energy equation
p = p(ρ)
equation of state
with ρ and p now assumed to include the cosmological constant term.
Note, we have 3 functions S (t), ρ(t) and p(t) to be determined, for which we need
3 equations to uniquely determine them in terms of some initial values.
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Cosmology
A) Empty Worlds, vacuum energy at most: ρ = p = 0 , Λ = κ ρvac
In spite of the fact that such models are unrealistic, they give some insight into the
geometrical and kinematic possibilities in cosmological model building. In this
limiting case we have the following basic equations:
Ṡ 2
c2
=
Λ
3
S2 − k ;
S̈
c2
=
Λ
3
S
or
Λ = −3
H 2 (t)
c2
q(t) ; K(t) =
H 2 (t)
− c2
(q(t) + 1) .
The general solution is
S (t) = a ct + b ; Λ = 0 , a2 = −k ,
Λ Λ
1 αct
−αct
S (t) =
ae + be
; Λ , 0 , α2 = ;
ab = k = (1, 0, −1) .
2
3
3
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Cosmology
Because of time reversal symmetry or anti-symmetry:
when b , 0: b = ±a with a > 0.
Cases:
(a) Λ = 0 , k = 0: S = S 0 = constant ; H = 0 , q = 0 .
(b) Λ = 0 , k = −1: S = ct ; H = 1/t , q = 0 ; z = t0/t1 − 1 ,
Milne’s Universe
(c) Λ > 0 , k = 1: S = a cosh
ct
a
.
(d) Λ > 0 , k = 0: S = exp
= exp (Ht) ; H ≡ = κ ρvac c2/3
de Sitter Universe: here H(t) = constant and q(t) ≡ −1, this universe has infinite
age! Today’s pure dark energy universe!
ct
a
(e) Λ > 0 , k = −1: S = a sinh
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a
ct
a
p
.
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Cosmology
(f) Λ < 0 , k = −1: S = a sin
ct
a
.
where in all cases
a=
p
|3/Λ| .
Models (b), (d), (e) and (f) some times have been called “incomplete” as they
exhibit more space than “substrate”.
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Cosmology
Λ > 0 , k = 0, ±1
Λ < 0 , k = −1
Terminology:
(ρ = p = 0)
(ρ = p = 0)
de Sitter spaces
Anti-de Sitter spaces
2 2 2 2
2
0
1
2
3
Isometric embedding: ds = dz − dz − dz − dz − k dz4
2
S 4−
6
z
0
z4
S 4+
60
z
z4
3
z3
z-
de Sitter
c 2009, F. Jegerlehner
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Anti de Sitter
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Cosmology
B) Matter dominated: ρ > 0 , p = 0
Λ = 0: preferable class of models
satisfies Einstein’s boundary criterion: empty space = flat space
Base equations: ρ S 3 = ρ0 S 03 = constant
Ṡ 2
c2
= 3κ ρ S 2 − k =
C
S
− k ; C 3κ ρ S 3 = constant
√
which can be integrated elementary: dS / C/S − k = ±c dt
v
k = 0 : Einstein-de Sitter universe
S (t) =
v
1/3
9
4C
(ct)2/3
k = 1 : Friedmann-Einstein universe
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Cosmology
ct = C arcsin
√
√
X−
X−
X2
; X = S (t)/C
Parameter representation:
=
=
S (t)
ct
v
1
2C
1
2C
(1 − cos η)
(η − sin η)
)
cycloid
k = −1 :
ct = C
√
X+
X2
√ − arcsinh X ; X = S (t)/C
Parameter representation:
S (t)
ct
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=
=
1
2C
1
2C
(cosh η − 1)
(sinh η − η)
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Cosmology
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Cosmology
In each case we have S (t) → 0 (t → 0) i.e., all of them are Big-Bang cosmologies.
For small t the C term is dominating, such that
S (t) '
1/3
9
4C
(ct)2/3 ; t → 0
For the observables we have the relationships
κ
2
ρ(t)
K(t)
=
=
H 2 (t)
c2
3
H 2 (t)
c2
q(t)
(2 q(t) − 1)
q(t) > 0!



>



=
q(t) = 



 <
1
2
1
2
1
2
;
;
;
k=1
k=0
k = −1
The following graphical representation is shown for C = 1 and c = 1, i.e, S (t) as
well as ct are represented in units of C .
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Cosmology
q = 0.10
0.16
0.21
∞
0.25
0.30
7.5
2.0
1.0
0.5
↑
1
ctc
2
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↑
ctc = π
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Cosmology
Einstein-de Sitter universe
The flat case k = 0 is of particular interest as a critical universe in between positive
and negative curvature. In this case many things are fixed:
First
K(t) ≡ 0 3 H 2(t)
1
.
qEdS(t) ≡ and ρEdS(t) =
2
2
κ c
In addition, using S 3 = 49 C(ct)2, we obtain
κ
κ 3 H 2(t) 3 9 H 2(t)
3
2
(ct)
C
C = ρS = ·
·
S
=
3
3 κ c2
4
c2
where C cancels and we get 1 = 49 H 2(t) (t)2 or
t = 32 H −1(t) (EdS)
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Cosmology
Hence in the EdS universe everything is fixed for given H0:
Present values:
H0
ρ0 EdS
t0 EdS
0.83 × 10−28 cm−1 × c
1.2 × 10−29 gr/cm3 × c2
2 −1
9
3 H0 ' 8.1 × 10 years (EdS age of the universe)
'
'
=
where we used
κ=
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8πG N
c2
= 1.86637 × 10−27 cm/gr .
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Cosmology
a(t)
t0
tH = H−1
0
t
Einstein-de Sitter universe: a typical expansion pattern (for closed universes only
for t trecontraction)
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Cosmology
Back to the non-flat geometries k = ±1:
here we have
κ
H 2(t)
3
C = ρ(t) S (t) = 2 2 q(t) S (t)3
3
c
thus, with
k S (t)−2
H 2(t)
= 2 (2 q(t) − 1) ,
c
we may write
S (t) =
c
H(t)
in terms of the observables H(t) and q(t).
1
|2 q(t)−1|
√
Furthermore, we have
c
q(t)
C=2
; q(t) ≥ 0 !
H(t) (|2 q(t) − 1|)3/2
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Cosmology
Also, with
Ṡ 2 C
S̈
C
S̈ S 1 C
= −k , 2 =− 2 ; q=− 2 =
2 C − kS
c2 S
c
2S
Ṡ
we obtain 1
k S (t) = C
2 q(t) − 1
,
2 q(t)
which means that kS (t) is determined by C and q(t).
1 We also may write
(
)
(√
)
√
q
q
2q−1
1−2q
k=−1
ct = C arcsin 1 − 2q−1 − 2q
; ct = C
2q−1 − 1
2q − arcsinh
k=1
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Cosmology
The conclusion of the above discussion: provided Λ = 0 and p = 0, we may
determine the age and the curvature of the universe by evaluating the observable
relations at t = t0. Since, in principle, we can determine H0 and ρ0 independently,
we can find whether k = 0, +1 or − 1. The density ρEdS for k = 0, shows up as the
critical density. Generally,
ρ(t) = ρEdS(t) · 2 q(t)
such that
ρ0 > ρ0 EdS
ρ0 < ρ0 EdS
k=1
k = −1
Thus:
l Lot of matter – space closes under gravity, gravity wins.
l Little matter – space is open and matter spreads forever.
i.e, either confinement or asymptotic freedom
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Cosmology
In any case the matter dominated energy density ρmat is known not to be given by
the normal baryonic matter which stars, planets, dust and interstellar gas are
made of. There are many indications that baryonic matter is only a fraction of the
total gravitating matter, which is mainly Dark Matter (DM). Dark matter, which only
manifests to us by gravitational interaction, is seen in galaxies (velocity profiles), in
relative motion of the objects in clusters of galaxies (Fritz Zwicky 1933, applying
the virial theorem to Coma Cluster) as well as in the universe as a whole (CMB
fluctuations). A very important “tool” in dark matter search is gravitational lensing,
which convincingly supports the other findings, and in addition provides important
information concerning the distribution of DM. We will discussed this in detail later,
together with Observational findings.
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C) Radiation dominated: p = ρ/3
Realistic for the early stage of a Big-Bang cosmology.
Λ = 0: preferable class of models
satisfies Einstein’s boundary criterion: empty space = flat space
Base equations: ρ S 4 = ρ0 S 04 = constant
Ṡ 2
c2
= 3κ ρ S 2 − k =
C̄
S2
which can be integrated elementary.
− k ; C̄ 3κ ρ S 4 = constant
We actually have
√ S dS
C̄−kS 2
v
= ±cdt
k=0:
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Cosmology
S (t) = 4 C̄
v
1/4
(ct)1/2
k=1:
√
21/2
S (t) = C̄ − C̄ − ct
√
in 0 < ct < 2 C̄ with cyclic extension !
v
k = −1 :
1/2
√
2
S (t) =
C̄ + ct − C̄
In each case we have S (t) → 0 (t → 0) i.e., all of them are Big-Bang cosmologies.
As in the matter dominated scenario, for small t the C̄ term is dominating, such
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that, independently of k,
S (t) ' 4 C̄
1/4
(ct)1/2 ; t → 0
The behavior for large t is easily obtained:
k=0
k=-1
k=1
:
:
:
S (t) = 4 C̄
S (t) ' ct
1/4
(ct)1/2
√
cyclic, cycle time ctc = 2 C̄
For k = 1 cyclicity is t → t − t0n ; t0n = n ·
√
2 C̄
c
.
For the observables we have the relationships
c 2009, F. Jegerlehner
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Cosmology
κ ρ(t)
K(t)
=
3
=
H 2 (t)
c2
H 2 (t)
c2
q(t)
(q(t) − 1)
q(t) ≥ 0!


>1;



=1;
q(t) = 


 <1;
k=1
k=0
k = −1
The following graphical representation√is shown for C̄ = 1 and c = 1, i.e, S (t) as
well as ct are represented in units of C̄ .
c 2009, F. Jegerlehner
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Cosmology
q = 0.10
0.2
0.3
0.5
ր 6.0 ∞
3.0
1.0
↑
ctc
1
2
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Lect. 7 x≫
↑ √
ctc = 2 C
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Cosmology
Actually, in the radiation dominated era the situation is qualitatively very similar to
the matter dominated p = 0 scenario.
Exercise: Express q(t) in terms of S and C (matter dominated) or C̄ (radiation
dominated) scenario, respectively.
For radiation in thermal equilibrium (black body radiation) the Stefan-Boltzmann
law holds:
ρradiation = a T c ; a =
4 2
4
π 2 kB
15 ~3 c2
= 8.418 × 10−36 gr/cm3 ◦K−4
and the spectral distribution is given by the Planck-distribution
c 2009, F. Jegerlehner
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Cosmology
ργ (ν) dν =
3
8π hν dν exp k hνT −1
B γ
which characterizes the perfect radiator.
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Cosmology
Since ρ(t)S (t)4 = ρ0S 04 = constant we have
1
ρ(t) ∝ T (t) ∝
S (t)4
4
implying
T (t) = T 0 SS(t)0
In fact, the radiation cools down by the expansion so much that it decouples from
matter. This was predicted by Gamow and Alpher and Herman in 1948,
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Cosmology
and Dicke and Peebles at Princeton were searching for it when it was actually
discovered “by accident” in 1965 by Penzias and Wilson at Bell Laboratories.
For more history read:R≫R≫
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Cosmology
The discovery of the isotropic Cosmic Microwave Background (CMB) of
temperature
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Cosmology
T 0 = (2.725 ± 0.002) ◦K
not only clearly favored Big-Bang cosmologies and essentially ruled out
steady-state theory, it by now provides the best test of isotropy at the level of 0.1%
over 360◦ of the sky! We will come to that later.
What does it mean? First we have
ρ0γ = a T 04 c2 ρ0γ ' 4.64 × 10−34 gr/cm3 × c2 , n0γ ' 410 photons/cm3
which confronts with the present baryonic matter density
ρ0,mat ' 3 × 10−31 gr/cm3 × c2 .
However, the evolution of the two densities are different
ρrad(t) = ρ0,rad
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S0
S (t)
!4
whereas
ρmat = ρ0,mat
S0
S (t)
!3
,
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Cosmology
and since S (t) → 0 (t → 0) in a Big-Bang cosmology radiation always dominates
in the early universe!
ρ(t)
ρrad (t)
radiation
era
present
Big Bang
ρrad = ρmat
matter
era
ρmat (t)
curvature ?
t
Radiation dominates the Big Bang, at present it is a cold relict the CMB
Besides photons, a similar sea of cold neutrinos fills our universe. This we will
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discuss later.
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The critical energy density and the flatness problem
In general we have some mixture of vacuum energy, relativistic and non-relativistic
matter. We have seen that the Einstein-de Sitter universe with k=0 plays a special
role. In the present matter dominated era ρEdS plays the boundary case between
expansion forever or recontraction. We therefore define
ρ0,crit = ρEdS =
3H02
8πG N
= 1.878 × 10−29 h2 gr/cm3,
where H0 is the present Hubble constant, and h its value in units of 100 km s−1
Mpc−1, and express the energy density in units of ρ0,crit. Thus the present
density ρ0 is represented by
Ω0 = ρ0/ρ0,crit
and
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Cosmology
Ω0 = 1
is the critical density for which the universe just remains open, infinite and flat.
Ω0 > 1
the case of much matter, space closes due to gravitational attraction of mass.
Gravity stops the expansion after finite time, the universe collapses, ends in heat
death!
Ω0 < 1
the case of little matter, gravity not sufficient to stop expansion, universe expands
forever, space open, ends in freezing to death!
If we include the cosmological constant as a vacuum energy density in the total
density ρ, we may write the Friedmann equations in the form
Ṡ 2
c2
+ k = 3κ ρ S 2 and
3 c2S̈S = − 2κ (3 p + ρ)
and the energy-momentum conservation as
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Cosmology
ρ̇ = −3 SṠ (ρ + p) .
If the mixture of radiation and matter dominates over the vacuum energy 3p + ρ is
always positive and thus we have S̈ /S ≤ 0. This means that the expansion must
have started with S = 0 in the past and the present age must be lower than the
Hubble age: t0 < H0−1 (see figure in EdS case above). For k = 1 this also implies
the recontraction to S = 0.
The present deceleration parameter q0 = −S̈ 0/
q0 =
κ (ρ0 +3 p0 )
6 H02
=
S 0 H02
can be written as
ρ0 +3 p0
2 ρ0,crit
This equation provides a simple explanation for the different values q0 takes
depending on the form of the equation of state:
c 2009, F. Jegerlehner
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Cosmology
Form of energy
equation of state q0
q0(k = 0)
a) vacuum energy:
pΛ = −ρΛ
q0 = −ΩΛ
q0 = −1,
i.e.
b) non-relativistic matter:
pmat = 0
q0 = 12 Ωmat q0 = 12 ,
c) relativistic matter, radiation: prad = ρrad/3
q0 = Ωrad
q0 = 1,
the different forms of matter contribute with different signs and weights to q0.
Whatever the mix of energies the total energy density

!3
!4



S0 
S0


ρ(t) = ρ0,crit 
+
Ω
Ω
+
Ω

0,rad
0,mat

 Λ

S (t)
S (t) 
for t → 0 is dominated in any case by the radiation part:
ρtot ' ρ0,crit Ω0,rad
c 2009, F. Jegerlehner
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Lect. 7 x≫
S0
S (t)
!4
,
34
Cosmology
in which case, independent of k!
S (t) ' 4C̄
1/4
(ct)
1/2
S 04
κ
3
4
; C̄ = ρ0,rad S 0 i.e.
,
=
3
S 4(t) 4 κ (ct)2 ρ0,rad
such that
ρtot(t) '
3
4 κ (ct)2
is universal.
Now, in the radiation dominated early phase after the Big Bang
Ṡ 2
c2
= 3κ ρ S 2 − k =
C̄
S2
− k ; C̄ 3κ ρ S 4 = constant
shows that the curvature term proportional to k is subleading and may be dropped
c 2009, F. Jegerlehner
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Cosmology
(in accord with the universal behavior just mentioned before):
Ṡ 2
C̄
Ṡ 2
c2 C̄ c2 κρ
→ 2 or
→ 4 =
3
c2
S
S2
S
i.e.
ρ(t) = ρEdS(t) =
3 H 2 (t)
κ c2 .
It is truly remarkable that at that early times, precisely when we would expect
curvature the be most important, the evolution automatically picks the flat solution.
This does not mean, however, that at later times when the other energy density
components come in to play and even dominate the scene we have to expect a flat
space one.
In fact the observed energy density at present is close to the critical one. This we
will discuss later. Even if one counts the normal baryonic matter only one is not
much more than a factor of 10 off from the critical value Ω = 1 (atoms ∼ 4%) . The
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Cosmology
missing part making Ω = 1 complete is dark matter (cosmological constant 73%)
and cold dark matter (what is it ? 23%). The strongly time and scenario
dependent energy density ρ(t) easily deviates by 60 orders of magnitude from
ρ0,crit today, given the evolution during the enormous time span of the present age
of the universe. In other words, the flat solution Ω = 1 is highly unstable. How is it
possible then that we apparently live in a universe close to that unstable point?
This is the so called flatness problem (Dicke 1969). The solution: either
cosmological fine-tuning assuming we really started with an accuracy of 62 or
more digits with Ω = 1 or some dynamical mechanism transmutes this unstable
point into an stable attractor, this is what inflation theory, to be discussed later, can
do for us.
Exercise: confirm the necessary fine tuning in the cosmic evolution by explicit
calculation.
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Cosmology
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Cosmology
Appendix: Λ , 0
The issue about the true value of the cosmological constant, which likely is
non-vanishing as mentioned earlier, will be reconsidered later. In any case we
have to discuss the solution of the cosmological equations for the case on
non-zero cosmological constant. The consequences for the empty world case we
already discussed: flat “background space” gets replaced by de Sitter or anti-de
Sitter space. The generalization to the matter dominated scenario is almost trivial.
A glimpse at the Friedmann equation shows that one can treat the cosmological
constant as a contribution to the energy density:
κρmat + Λ → κρtot,
where ρtot = ρtot + ρΛ with ρΛ = Λκ . Provided p = 0, all solutions remain unchanged,
except for a different interpretation of the energy density, which in this case is not
identical with the “normal” mass density of baryonic plus dark matter. Note that
the cosmological constant in general enters the equation of state in a not a priori
known way. However, this seems not to matter as long as we have p ' 0.
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Cosmology
That a non-vanishing Λ spoils the geometry ⇔ matter duality, because the
Einstein equation remains true no matter on which side of the equation we write
the cosmological term. Many believe it has some thing to do with quantum
vacuum fluctuations, but no answer can be given why it is so small. In any case if
we take the energy momentum tensor as given on the quantum level by the
Standard Model (SM) of strong and electroweak interactions we would expect the
Higgs field vacuum expectation value (Bose condensate) as well as the quark
condensates to contribute to the cosmological constant. Again, the value obtained
is about 50 orders of magnitude to big! What tames the cosmological constant to
a small non-vanishing value?
Exercise: estimate the cosmological constant as induced by the Higgs mechanism
and by spontaneous symmetry breaking of chiral symmetry in QCD.
A cosmological constant of course affects local gravitation in particular it might
affect black holes by changing the Schwarzschild radius etc. In order to study the
physical effect of a non-vanishing Λ we consider the spherically symmetric mass
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Cosmology
distribution in outer space:
Gµν ≡ 0 Gµν = Λ gµν
Ê Λ = 0 : Schwarzschild
1
r
0
2
2
2
(cdt)2 −
dr
−
r
dΩ
ds2 = 1 −
r
1 − rr0
which is unique, static with boundary condition: flat as r → ∞. It has the Newtor
2ϕ
nian approximation with potential: ϕ ' − mr with m 20 and where g00 ' 1 + c2
Ë Λ,0:
c 2009, F. Jegerlehner
≪x
Lect. 7 x≫
41
Cosmology
α= 1−
r0
r
→ 1 − rr0 − Λ3 R2. Denoting K =
Λ
3
r
1
0
2
2
2
ds2 = 1 − − Kr2 (cdt)2 −
dr
−
r
dΩ
r0
r
1 − r − Kr2
which is unique, static with boundary condition: Schwarzschild as Λ → 0. In the
Newtonian approximation: ϕ ' − mr − 12 kr2
Observation of planetary motions yields: |Λ| 10−42 cm−2 !
In particular: in empty space (no matter) m =
Λ > 0. The gravity potential is
r0
2
= 0 we have de Sitter space if
1
ϕ ' − Kr2 repulsive linear force F ∝ Kr
2
c 2009, F. Jegerlehner
≪x
Lect. 7 x≫
42
Cosmology
and the metric
ds = 1 − Kr
2
2
1
2
2
2
(cdt) −
dr
−
r
dΩ
1 − Kr2
2
which is spherical symmetric with respect to any point and is regular. For K > 0
there is the de Sitter coordinate singularity K r2 = 1. While for m , 0 we have a
true singularity at r = 0, such a singularity is absent for m = 0.
Mappings
Λ > 0: de Sitter space dS 4
embedding into M 5 = R1,4 : (cT )2 − X 2 − Y 2 − Z 2 − W 2 = −a2
ds = (cdT ) − dX + dY + dZ + dW
2
c 2009, F. Jegerlehner
≪x
Lect. 7 x≫
2
2
2
2
2
43
Cosmology
With a2 1
K
=
3
Λ
; a > 0 the mapping reads:
X
= r sin θ sin ϕ
Y
= r sin θ cos ϕ
Z
W
T
W
T
= r cos θ
q
2
= a 1 − ar 2
q
2
= a 1 − ar 2
q
2
= a ar 2 − 1
q
2
= a ar 2 − 1
cosh cta
sinh cta
sinh cta
cosh cta






r<a











r>a.





Note: z = constant is the hyperboloid: (cT )2 − W 2 = constant. Homogeneity and
Isotropy follow from the symmetry: S 4− is invariant under 5-dimensional Lorentz
transformations. The force is linear repulsive F ∝ 13 Λ r.
c 2009, F. Jegerlehner
≪x
Lect. 7 x≫
44
Cosmology
Λ < 0: Anti-de Sitter space AdS 4
embedding into R2,3 : (cT )2 − X 2 − Y 2 − Z 2 + W 2 = a2
ds = (cdT ) − dX + dY + dZ
2
2
With a2 1
K
2
2
2
+ dW 2
= − Λ3 ; a > 0 the mapping reads:
X
= r sin θ sin ϕ
Y
= r sin θ cos ϕ
Z
= r cos θ
r
W
T
c 2009, F. Jegerlehner
≪x
Lect. 7 x≫
r2
ct
= a 1 + 2 cos
a
a
r
r2
ct
= a 1 + 2 sin
a
a
45
Cosmology
Note: z = constant is the circle: (cT )2 + W 2 = constant. Homogeneity and Isotropy
follow from the symmetry: S 4+ is invariant under S O(2, 3). The force is linear
attractive F ∝ − 13 Λ r. Causality problem: periodicity in time ct → ct + 2 π a. The
circle in the (cT, W) plane is acausal. May be cured by mapping the circle to the
real line: which is the universal cover of the AdS space.
For the cosmological solutions of the Einstein-Friedmann equations the RW-metric
is only affected as far as S (t) solves a different dynamical equation.
Exercise: find the model Einstein proposed: an eternal static solution. Show that
Einstein’s GRT with vanishing cosmological constant has no such solution.
Exercise: the observation of planetary motions constrains the cosmological
constant to |Λ| 10−42 cm−2 !. Show that this is compatible with a dark energy
density, which has been determined to be ΩΛ ∼ 0.74 ± 0.03.
Previous ≪x , next x≫ lecture.
c 2009, F. Jegerlehner
≪x
Lect. 7 x≫
46