Download Inflation and the cosmological constant problem

Inflation and
the cosmological constant
problem
Larissa Lorenz – Sebastian Sapeta
Krzyzowa 18.−28. September 2002
Contents
• Standard model of cosmology
and its problems
• The inflationary paradigm
• Review of the most important
inflationary scenarios
• Cosmological constant and
inflation
• Experimental determination of
cosmological constant
• The cosmological constant
problem
Big
Bang
N
S
φ0
φ(T) = φ0
φ(T) = 0
φ(T) = φ0φ(T) = φ0
About 15 billion years ago, time and space both began in a cosmological
singularity.
The standard model of cosmology
t=0
t = 10−43s
Big
ρ=
t = 10−10s
t = 1 billion years
MP
g
94
=
10
l P3
cm3
á!
il
Vo
t = 10−35s
Bang
−infinitely high temperature T
−infinitely large density ρ
−average particle energy exceeds
MPc2 = 1019GeV
quantum gravitation
era of unity of all interactions
except for gravity
Experimental evidence:
• the expansion of the universe
• the abundance of elements
• the cosmic microwave background
−baryon creation
era of unity of electromagnetic
and weak interaction
symmetry breaking between
electromagnetic and weak
interaction
−galaxy and star formation
Problems:
¥ the flatness problem
¥ the horizon problem
¥ the monopole problem
¥ the uniqueness problem
Why was the early universe so extremely flat?
The flatness problem
Let ρc be the critical density of the universe and ρ0 its present day value.
For the parameter Ω0 = ρ0/ρc, we assume
(1)
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The development of the deviation from the critical value |Ω − 1| in the course of time is given by
for the radiation dominated universe
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| Ω − 1 |~ t
3
2
for the matter dominated universe
In a radiation dominated universe of the age of t0 = 1017s, (1) leads to
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| Ω − 1 | < 10
− 16
| Ω − 1 |< 10 − 28
| Ω−1|< 10−50
Extremely flat!
today
nucleosynthesis
electro−weak
symmetry breaking
10−34s
Why is the 3K radiation so highly isotropic?
The horizon problem
Any observer‘s event horizon is given by
t
d
H
(t) = R (t)∫
0
dt ′
R ( t ′)
(2)
Therefore, two observer separated by s ≥ 2dH(t) are completely independent at any
time.
dH
dH
s
Owen
Let‘s assume that 10−35s after Big Bang the temperature T was approx. 1015GeV and
that R ~ 1/T; using the present day value of R and (2) , we find that today s = 2.4m!
Oscar
We should be observing
different causally
independent regions!
The microwave background
radiation is isotropic at a
scale of 1028cm (thermal
equilibrium)!
Where have all the monopoles1) gone?
The monopole problem
All Grand Unified Theories predict superheavy stable particles carying magnetic charge.
mm = 1016 mproton
S
These magnetic monopoles should be
as abundant as protons.
N
This means that the density in the universe at present should
be
1015 times higher that observed!
predicted
1) and other relics
g
ρ =10
cm3
−14
g
ρ =10
cm3
−29
observed
“What I am really interested in is
whether God could have created the world differently.” – A. Einstein
The uniqueness problem
Elementary particle theories turn out to present many different, yet equivalent solutions.
So why looks low energy physics around as the way it does – and not completely
different?
Why is space−time four−dimensional?
134
136
135
138
137
Why has the fine−structure constant this value?
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Slightly different values of physical constants would have led to a totally different
universe.
The inflationary paradigm provides answers to the standard model’s problems.
Problems and „inflationary“ answers
In classical big bang theory, |Ω−1| moves away from 0 in the course of time,
The flatness
problem
but during the inflationary period we find |Ω−1| approaching 0. If it gets close
enough, it stays close during all of the post−inflationary period.
No need for the early universe to be extremely flat!
The inflationary paradigm provides answers to the standard model’s problems.
Problems and „inflationary“ answers
A small area in the very early universe which reached thermal equilibrium
could today be larger than the oberserved universe due to the dramatical
increase during the inflationary period.
The horizon
problem
Thermal equilibrium is the simplest and correct explanation for
the isotropy of 3K radiation!
today observed universe
time
INFLATION
small area
The inflationary paradigm provides answers to the standard model’s problems.
Problems and „inflationary“ answers
The monopole
problem
While the universe is expanding exponentially, the monopole densitiy
decreases much faster than the energy density.
Monopole density in the universe can be neglected!
The uniqueness
problem
Maybe the reason why α equals 1/137 is that if it was otherwise, our kind of
life could not exist in the universe.
The metastabile, supercooled state of the universe causes inflation.
Guth‘s inflationary scenario
Let φ be a scalar field with its minima for
T > Tc
(superhigh
temperature)
T > Tc
T < Tc at
φ(T) = 0
φ(T) = 0, φ(T) = φ0
T < Tc
φ(T) = φ0
φ(T) = 0
phase transition from
φ(T) = 0 into ground
state φ(T) = φ0 via
new ground state
restored symmetry,
ground state
The universe remains in this
metastabile (supercooled) state.
The universe expands.
This leads to exponential
expansion.
Problems:
¥collision of bubble walls destructs homogeneity and isotropy
¥supercooled state has to be “stable” to create enough inflation
bubble formation
φ0
φ(T) = φ0
φ(T) = 0
φ(T) = φ0φ(T) = φ0
The universe gets hot
and can be described by
standard big bang
cosmology.
Inflation can not only take place in a supercooled state, but also during the process of φ
approaching the value φ0.
The new inflationary universe scenario
barrier
height ~ T4
minimum of
potential V(φ)
T = 1015 GeV
φ=T
Coleman−Weinberg potential
φ‘‘+3Hφ‘+V‘(φ)=0
T < 109 GeV:
tunnelling probality
reaches 100%
At about T < 109 GeV, the
barrier becomcs
unstable.
Oscillations around the
minimum are damped by
expansion and the production of
relativistic particles.
This part has to be flat
enough to ensure a
sufficiently long time of
“rolling downhill” = time of
inflation (much longer than H−
1).
Problems:
• elementary particle theory required whose effective potential satifies
many „unnatural“ constraints
• start of inflation depends on drop of temperature which takes 6 orders of
magnitude longer than Planck time
Thermalzation leads to
temparatures like at the
beginning of inflation (T < 109
GeV); future development is
described by the standard big
bang model.
Instead of assuming a certain value,
the scalar field’s φ initial distribution is regarded as chaotic.
The evolution of a scalar field
The chaotic inflation
..
.
φ with mass m
<< MPl is described by the Klein−Gordon equation:φ + 3H φ = −m 2φ
scenario
(3)
In the chaotic inflation scenario, no specific initial value is assumed but a chaotic distribution of values of φ0. If the
initial value of the field φ0 exceeds 1/5 MPl, the friction term is big enough to make the solution rapidly approach
ϕ ( t) = ϕ 0 −
the regime:
for
t<
mM Pl
2 3π
and
R(t ) = R0 exp(
MP4−Planck density
φ
An island of classical
space−time rises out
of the space−time
foam; typical initial
value of φ is:
2
M Pl
m
(4)
ϕ0
4π mφ (t)
⇒ R(t) = R0 exp(
t ) = R0 exp( H (ϕ )t)
mM Pl
3 M Pl
V(φ)
ϕ0 =
t
Inflation takes
places while the
field is rolling
downhill with
friction; typical
expansion is
1010^9!
Inflation stops when φ
reaches its minimum,
friction becomes negligible
and φ performs oscillations
round the minimum;
energy is used for particle
creation.
2π
[ϕ02 − ϕ 2 (t )])
2
M Pl
(6)
(5)
quasi−exponential expansion !
This scenario offers some amazing
features:
Processes separated by at least H –1 are
completely independent.
⇓
Any inflationary domain of initial size exceeding
2H –1 can be considered as a separate mini−
universe!
That means, a region of size H –1 which
expands exponentially during the period of
inflation creates a lot of new mini−universes.
Of those, exponentially many have smaller
than
the
“mother”
universe,
and
φ0
also
Only this model of inflation can explain why the observable part of the universe is so homogeneous,
but from it also follows
exponentially many have bigger φ0.
that on a much larger scale the universe is extremely inhomogeneous. Moreover, realistic models of elementary particles
consider many kinds of scalar fields whose potential energy may have several different minima. So…
⇒ The Universe is divided into domains with various laws of particles physics or even dimensionality.
Cosmological constant may be identified with a vacuum energy density.
Why do we bother with Λ ?
Originally Introduced by Einstein as a free parameter to the field equations to
balance an attractive gravitational force and to allow a static universe.
1
Rµν − Rg µν + Λg µν = 8πGTµν
2
(7)
The idea of Λ came back in the context of modern quantum field theories in which
the vacuum is not necessarily a state of zero energy but it is defined as a state of
the lowest energy.
Due to the Lorentz invariance of the ground state the vacuum energy momentum tensor has to be
proportional to
.
(8)
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The effect of an energy−momentum tensor of the form (8) is equivalent to that of a cosmological
constant from (7) and this is the origin of the identification of the cosmological constant with the
energy of the vacuum.
Λ
ρ vac = ρV =
(9)
8πG
The vacuum can therefore be thought of as a perfect fluid with the equation of state:
(10)
There exist three different contributions to Λ:
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• the static cosmological constant Λgeo
• quantum fluctuations Λfluc
• additional contributions due to currently unknown particles and interactions Λinv
(11)
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Non−vanishing Λ may be responsible for inflationary expansion.
Cosmological constant and inflation.
Under the assumption of an homogeneous isotropic universe and non−vanishing cosmological
constant Einstein equations can be reduced into Friedmann equations:
2
 R. 
8πG
k Λ
H 2 ≡   =
ρ− 2 +
3
R
3
R
 
time
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..
4πG
R
=−
(ρ + 3 p )+ Λ
3
3
R
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First two terms in (12) decrease quickly in the expanding universe
while the third remains constant. At last cosmological constant
begins to dominate and we can write:
2
 R. 
Λ

H 2 ≡   =
3
R
 
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This leads to the exponential expansion:
 Λ 
R(t ) ∝ exp
t 
3


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How does the history of the universe depend on Λ value ?
Model universes and their fates
The Friedmann equation:
A positive cosmological constant accelerates the expansion,
..
R
4πG
(ρ + 3 p )+ Λ
=−
R
3
3
while a negative Λ and ordinary matter decelerate it.
Λ c = 4(8πGM )
−2
− the value of Einstein’s static universe
R(t)
R(t)
0<Λ<Λc,k=1
Λ<0,k=−1,0,+1
Λ=0,k=1
R(t)
Λ=0,k=−1
t
t
Λ>0,k= 0,−
R(t)
Λ=Λc(1+ε),k=1
1 Λ>Λc,k=1
ε <<1
Λ=0,k= 0
t
t
Critical density and deceleration parameter.
Alternative notation for Friedmann equations.
Friedmann equation:
2
 R.  8πG
k Λ
 
H2 ≡   =
ρ− 2 +
3
3
R
R
 
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(16)
may be write in a form:
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where
and
ΩM =
ρ
ρc
Ωk = −
k
a2 H 2
ΩΛ =
Λ
3H 2
ρc− critical density of matter in the universe
The universe is flat provided that ΩM +ΩΛ = 1
Let us introduce a deceleration parameter:
&
&R 1
R
q ≡ − 2 = Ω M − ΩΛ
R& 2
(17)
Positive q cause the universe to decelerate while negative q leads to
acceleration
Observational tests implies (ΩM=0.3 , ΩΛ=0.7)
Determination of Λ using type Ia supernovae
The luminosity distance is defined as:
d l2 =
L
4πF
(18)
where L is the luminosity and F the measured flux of the galaxy
One can show that:
H 0d l = z +
1
(1 − q0 )z 2 + ...
2
(19)
We can obtain the value of q0 by measuring the luminosity distance and red−shift.
The best method to determine dl is to use the “standard candle” properties of type Ia supernovae.
Observational test implies non vanishing Λ
Traditional model
of flat universe
without Λ is not
favoured. There
seems to be
strong evidence
for a positive Λ
and accelerating
universe.
Independent measurements suggest the universe with non−vanishing Λ.
Determining Λ by measuring angular diameter of objects.
The angular diameter distance is defined as:
(20)
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where D is the proper diameter of an object and Θ its apparent angular size
It can be shown that:
H 0d A = z −
1
(3 − q0 )z 2 + ...
2
(21)
Measuring the angular diameter of objects and the red−shift allows a determination of
q0
Data from 82 compact radio
sources results in evidence
for q0 of approximately 0.5.
There are many ways in which cosmological constant can manifest itself.
Alternatives to determine Λ.
Observations of numbers of galaxies as a function of red−shift
Counting of galaxies
are sensitive test of ΩΛ
Examination of 1000 infrared galaxies results in:
Ω0 = 0.9 +−00..75 → q0 = 0.45+−00..35
25
An accelerating universe seems to be trustworthy.
Alternatives to determine Λ.
The lens probability rises dramatically as ΩΛ is increased to
Gravitational lensing
unity as we keep Ω fixed.
The existing data allow us to place an upper limit on
ΩΛ < 0.7 in a flat universe
Determining of ΩM by “weighing” clusters of galaxies.
Alternatives to determine Λ.
Many cosmological tests constrain some combinations of ΩΛ
Matter density
and ΩM. It is useful to determine ΩM independently by adding
masses of clusters of galaxies.
Measurements imply:
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Estimated values of Λ are in total contradiction to reality
The Λ problem.
A relaivistic field can be considered as a sum of harmonic oscillators of all possible frequencies ω.
In the case of a scalar field with mass m, the vacuum energy is the sum of all contributions:
1
E0 = ∑ hω j
j 2
One can show that:
4
E0 k max
ρV = lim 3 =
L →∞ L
16π 2
(22)
(23)
Assuming the validity of the general theory of relativity up to the Planck scale kmax=
ρV ≈ 10 92 gcm−3
lPl we get:
while the experimental value is of order
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121 orders difference between experimental and theoretical value of ρV !!!
Quantum fluctuations exist due to virtual particle creation.
Let us assume that these particles take up for a short time their Compton volume Lc3.
h
m c 3m 4
→ ρV = 3 = 3
Lc =
mc
Lc
h
(24)
This should produce effects noticeable on scales of meters to kilometers while there
is no evidence for any effect of Λ at the distances of 1028 cm.
Effects of Λ should be observed in today known universe but they are not.
Several suggested solutions for the Λ problem.
Supersymmetry
In this theory for every boson there is fermion which is its supersymmetric partner.This special
symmetry is associated with a supercharges Qa.
Hamiltonian has a form:
}
(25)
Qα 0 = Qα+ 0 = 0
(26)
α
In a supersymetric state:
Which implies
{
H = ∑ Qα , Qα+
0H 0 =0
(27)
Contributions from bosons are canceled by contributions from fermions and the energy of the
vacuum state vanishes.
The problem is that supersymmetry seems to be broken in the observed world.
String theory
The search is on for a four−dimensional string theory with broken supersymmetry and
vanishing or very small cosmological constant.
Several suggested solutions for the Λ problem.
Feynman’s path integrals and the principle of least action
Using Feynman’s path integral formalism and the principle of least action one can get that the
wavefunction of the universe Ψ has a form:
Ψ ≈ e3π / h GΛ
(28)
If we consider Λ as a free parameter, it turns out that this expression has a prominent maximum
for Λ = 0, which would solve the cosmological constant problem.
Holographic theory (speculative)
The number of degrees of freedom in a region grows as the area of its boundary, rather than its
volume. Therefore the conventional computations of Λ involves a vast overcounting of degrees of
freedom.