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Dark Matter, Dark Energy, and
Unification Models

Neven Bilić
Ruđer Bošković Institute
Zagreb
Lecture 1
1
2
3
4
5

BIG BANG
History of the Early Universe - Universe Creation
History of the Early Universe – Inflation
Cosmological Observations and History of the Universe
Empirical Grounds for Observational Cosmology
5.1 Hubble’s Law
5.2 Cosmic Microwave Background (CMB) Radiation
5.3 Big Bang Nucleosynthesis
6 Theoretical Foundation of Modern Cosmology
7 Density of Matter in Space
8 Dark Matter
Lecture 2
9 Dark Energy
9.1 Vacuum Energy – Cosmological Constant
9.2 Quintessence
9.3 Phantom quintessence
9.4 k-essence
Lecture 3
10 DE/DM Unification – Quartessence
10.1 Chaplygin gas
10.2 Problems with Nonvanishing Sound Speed
10.2.1 Spherical Model
10.3 Generalized Chaplygin Gas
10.4 Tachyon Condensate
10.5 Entropy Perturbations
10.6 Dusty Dark Energy
11 Conclusions and Outlook
1. Big Bang
What kind of explosion is the Big Bang?
The Big Bang is like an explosion of a bomb that took place
in a previously empty space. Wrong!

The Big Bang is an eplosion of space itself which, before that, was
concentrated in one point. Right!

C.H. Lineweaver and T. Davis, Misconceptions About the BIG BANG, Scientific
American, March 2005
2. Early Universe - Creation
The Universe begins as a quantum fluctuation of the initial
vacuum
The total energy of the Universe (matter energy+dark energy
+gravity) = 0
Owing to the uncertainty relations DE Dt  h
h – Planck constant

Borrowing a small amount of energy (E0 ) from the
vacuum is allowed for a long time (t  )!
As a consequence, the lifetime of the Universe is
almost infinite
3. Early Universe - Inflation
A short period of inflation follows – very rapid
expansion 1025 times in 10-32 s.
The horizon problem.
The horizon problem.
Observations of the cosmic
microwave background radiation
(CMB) show that the Universe is
homogeneous and isotropic. The
problem arises because the
information about CMB radiation
arrive from distant regions of the
Universe which were not in a causal
contact at the moment when radiation
had been emitted – in contradiction
with the observational fact that the
measured temperature of radiation is
equal (up to the deviations of at most
about 10-5) in all directions of
observation.
The flatness problem
Observations of the average matter density,
expansion rate and fluctuations of the CMB
radiation show that the Universe is flat or with a
very small curvature today. In order to achieve
this, a “fine-tuning“ of the initial conditions
is needed, which is rather unnatural. The answer
is given by inflation:

v =5
105Hd ,

The initial density perturbations problem.
The question is how the initial deviations from
homogeneity of the density are formed having in mind that
they should be about 10-5 in order to yield today’s
structures (stars, galaxies, clusters). The answer is given
by inflation: perturbations of density are created as
quantum fluctuations of the inflaton field.
4. Cosmological Observations and
History of the Universe

•
•
•
•
•
•

The properties we would
like to find out:
Spacetime geometry
Age of the Universe
Expansion rate
Amount of ordinary
(“baryonic”) matter
Amount and nature of
dark matter and dark
energy
Mechanism of structure
formation

•

•
•

•
•
•

Observations that could
yield these properties:
Background microwave
radiation
Observations of distant
dalekih supernovae
Large scale structure
observations
Concentrations of light
elements
Galactic rotation curves
Gravitational lenses
13 billion years

Universe Today

5 billion yeras

Solar System Foramation,
First Supernove

1 billion years

Galaxy and Star Formation

100 000 years

100 s

10-10 s

10-34 s

Formation of Atoms,
Decupling of Matter–Radiation
Nucleosynthesis of Light Elements
Synthesis of Protons and Neutrons
QCD Plasma: quarks and gluons
Electroweak Transition
End of Inflation

10-43 s

Quantum Gravity Era
After a short period of inflation the Universe
continue expanding at much slower rate so that
after roughly 13.5 billion years, the visible part is
about 90 billion light years.
5. Empirical Grounds for Observational
Cosmology
• The expansion of the Universe - Hubble’s
law

v = Hd

• Cosmic microwave background radiation very homogeneous in all directions
• Big Bang nucleosynthesis - proportions of
light elements (H, D, He, Li)
Hubble’s law –
Universe expansion
Hubble’s law –
Universe expansion

a=0.5
Hubble’s law –
Universe expansion

a=0.75
Hubble’s law –
Universe expansion

a=1
Hubble’s law –
Universe expansion

recession
velocity

a=0.5

a=0.75
a=1
CAN GALAXIES RECEDE FASTER THAN LIGHT?
Of course not. Einstein’s special
theory of relativity forbids that.

Wrong!

Sure they can. Special relativity
does not apply to recession
velocity. Right!

RH

RH - Hubble radius
CAN WE SEE GALAXIES RECEDING FASTER
THAN LIGHT?
Of course not. Light from those
galaxies never reaches us.

Wrong!

Sure we can, because the expansion
rate changes over time. Right!
Do objects inside the Universe expand, too?
Yes. Expansion causes the universe and everything in it to grow.

Wrong!

Ne. No. The universe grows, but coherent objects (stars, galaxies ..) do not

Right!
T = 2.723K

DT = 100K

DT = 200K

Measuring CMB; the temperature map of the sky.
Angular (multipole) spectrum of the fluctuations of the CMB
(Planck 2013)
6. Theoretical Foundation of Modern
Cosmology
• General relativity – gravity

G=T
Matter determines the space-time geometry
Geometry determines the motion of matter
• Homogeneity and isotropy of space – approximate
property on very large scales (~Glyrs today)
• Fluctuations of matter and geometry in the early
Universe cause structure formation (stars, galaxys,
clusters ...)
General Relativity – Gravity

1
R  g  R  8 T  g  
2

g - metric tensor
R - Riemann curvature tensor
 - cosmological constant
T  - energy – momentum tensor
Perfect Fluid
Matter is described by a perfect fluid

T  (   p)u u  pg 
u - fluid velocity
T- energy-momentum tensor
p - pressure
ρ - energy density
The energy momentum conservation

T  ; = 0
yields, as its longitudinal part

uT  ; = 0 , the continuity equation


  3H (   p) = 0,
and, as its transverse part, the Euler equation


u =

1
h   p, ,
p

where

3H = u ; ;
h  = g   u u


 = u , ;


u  = u u  ; .

projector onto the three-space orthogonal to

H is the local Hubble parameter.

u
Cosmological principle
Homogeneity and isotropy of space
 dr 2
2
2
ds  dt  a (t ) 
 r d 
2
1  kr

2

2

2

- cosmological scale
the curvature constant k takes on the values 1, 0, or -1,
for a closed, flat, or open universe, respectively.

a (t ) – cosmological scale
Friedman equations

k 8 G
a
 2 

 
3
a a
4 G
  
a
(   3 p)
3
2

T  ;  0


a
H
a

k

expansion rate



 0 closed
 0 flat
 0 open

  a 3(1 w)

w p 

Various kinds of the cosmic fluids with different w
• radiation
• matter
• vacuum

pR   R 3
pM  0
p    

w 1 3
w0
w  1

  a 4
  a 3
  a0
7. Density of Matter in Space
The best agreement with cosmologic
observations are obtained by the models with a
flat space
According to Einstein’s theory, a flat space
universe requires critical matter density cr
today cr  10-29 g/cm3
Ω=  /

cr ratio of the actual to the critical density

For a flat space Ω=1
What does the Universe consist of?
From astronomical
observations:
luminous matter (stars,
galaxies, gas ...)
lum/cr0.5%

From the light element
abundances and comparison
with the Big Bang
nucleosynthesis:
barynic matter(protons,
neutrons, nuclei) Bar/cr5%

Total matter density fraction ΩM=M/cr 0.31
Accelerated expansion and comparison of the standard Big
Bang model with observations requires that the dark energy
density (vacuum energy) today ΩΛ=  /cr = 0. 69%
Density fractions of various kinds of matter today with
respect to the total density

B
B 
 0.05
 tot

DM

 DM

 0.26
 tot


 
 0.69
 tot

These fractions change with time but for a
spatialy flat Universe the following always holds:

 tot  crit
B  DM    1
Age of the Universe
Easy to calculate using the present observed fractions of
matter, radiation and vacuum energy.
For a spatially flat Universe from the Friedmann equation
and energy consrvation we have

H (a)  H 0 (  M a 3  Ra 4 )1/2

H 0  h  100Gpc/s2  (14.5942Gyr)1, h  0.67
Age of the Universe is then
T

1

1

da
1
da a
T   dt  

 (a 3  M )1/2
aH H 0 0
0
0

T  13.78Gyr
How Large is the Observable Universe?
S The universe is 14 billion years
old, so the radius of the
observable part is 14 billion lightyears. Wrong!

Because space is expanding, the
observable part of our Universe
has a radius of more than 14
billion light-years. Right!
8. Dark nature of the Universe – Dark
Matter

•
•
•
•

According to present
recent obsrvations
(Planck Satellite Mission):
More than 99% of matter
is not luminous
Out of that less then 5%
is ordinary (“baryonic”)
About 26% is Dark Matter
About 61% is Dark
Energy (Vacuum Energy)

0.69
0.26
0.04
0.01
DARK MATTER
Galactic rotation curves
Cluster 1E0657-558, Bullet

Clowe et al, astro-ph/0608407
http://chandra.harvard.edu
Neven Bilic, "Dark Matter, Dark Energy, and Unification Models"
Neven Bilic, "Dark Matter, Dark Energy, and Unification Models"
DM contents
- possible candidates Baryonic DM
in the form of
astrophysical objects
•
•
•
•

Brown dwarfs
Black holes
MACHO
planets

Nonbaryonic DM
• Sterile neutrino
• Axion
• SUSY stable particles:
gravitino, neutralino,
axino

Exsperimental evidence?
- great expectations from the LHC experiments at CERN
Hot DM refers to low-mass neutral particles that are still
relativistic when galaxy-size masses (  1012 M  ) are first
encompassed within the horizon. Hence, fluctuations on
galaxy scales are wiped out. Standard examples of hot
DM are neutrinos and majorons. They are still in
thermal equilibrium after the QCD deconfinement
transition, which took place at TQCD ≈ 150 MeV. Hot DM
particles have a cosmological number density
comparable with that of microwave background
photons, which implies an upper bound to their mass of
a few tens of eV.
Warm DM particles are just becoming nonrelativistic
when galaxy-size masses enter the horizon. Warm
DM particles interact much more weakly than
neutrinos. They decouple (i.e., their mean free path
first exceeds the horizon size) at T>>TQCD. As a
consequence, their mass is expected to be roughly
an order of magnitude larger, than hot DM particles.
Examples of warm DM are keV sterile neutrino,
axino, or gravitino in soft supersymmetry breaking
scenarios.
Cold DM particles are already nonrelativistic
6
when even globular cluster masses (  10 M )
enter the horizon. Hence, their free is of no
cosmological importance. In other words, all
cosmologically relevant fluctuations survive in
a universe dominated by cold DM. The two
main particle candidates for cold dark matter
are the lowest supersymmetric weakly
interacting massive particles (WIMPs) and the
axion.
9. Dark Energy
Because gravity acts as an attractive force
between astrophysical objects we expect that
the expansion of the Universe will slowly
decelerate.
However, recent observations indicate that the
Universe expansion began to accelerate since
about 5 billion years ago.
Repulsive gravity?
Lecture 2
Accelerated expansion   0
One possible explanation is the existence of a fluid
with negative pressure such that

p  3  0
and in the second Friedmann equation the universe

acceleration a becomes positive
cosmological constant   vacuum energy density
with equation of state p=-ρ. Its negative pressure may
be responsible for accelerated expansion!
New term: Dark Energy – fluid with negative
presssure - generalization of the concept of vacuum
energy
Accelerating universe?

Cosmological scale

accelerating

open

closed

today

time
Time dependence of the DE density
Another important property of DE is that its density does not
vary with time or very weakly depends on time. In contrast ,
the density of ordinary matter varies rapidly because of a
rapid volume expansion.
The raughh picture is that in the early Universe when the
density of matter exceeded the density of DE the Universe
expansion was slowing down. In the course of evolution the
matter density decreases and when the DE density began to
dominate, the Universe began to accelerate.
Most popular models of dark energy
• Cosmological constant – vacuum energy density.
Energy density does not vary with time.
• Quintessence – a scalar field with a canonical kinetic
term. Energy density varies with time.
• Phantom quintessence – a scalar field with a negative
kinetic term. Energy density varies with time.
• k-essence – a scalar field whose Lagrangian is a
general function of kinetic energy. Energy density varies
with time.
• Quartessence – a model of unifying of DE and DM.
Special subclass of k-essence. One of the popular
models is the so-called Chaplygin gas
9.1 Cosmological
Constant – Einstein’s
biggest mistake?
1917 Einstein proposes
cosmological constant
1929 Hubble discovers
expansion of the Universe
1934 Einstein declares the
cosmological constant his
biggest blunder
1998 Pearlmutter, Riess, and
Schmidt find evidence for the
cosmological constant or
dark energy
Origin of the cosmological constant:
Damn!
The vacuum is not
empty.

Quantum fluctuations
of the vacuum

W+
W-

e+

quark
eantiquark

particle
antiparticle
Observations of supernovae type Ia
Od 1988 Saul Perlmutter (Univ. Of
California) lieder of Supernova
Cosmology Project

Od 1994, Brian Schmidt (Australian
National University) klieder of High z
Supernova Search Team

In the same team Adam Riess (Space
Telescope Science Institute)
the first one who made the analysis
Supernovae Ia occure in binary systems made of a white
dwarf and a large companion star. A low-mass white dwarf
accreting matter from a nearby companion approaches the
limit of 1.4 solar masses (Chandrasekar limit) and becomes
unstable.
Then, a thermonuclear
explosion takes place
ensues and an
immense amount of
energy is suddenly
released.
In a typical galaxy,
supernovae occur a
few times
in thousand years.
In a short period a
huge amount of
energy is released
After a few days the
emission achieves
a maximum with
brightness
comparable with
the luminosity of an
entire galaxy.
Supernova SN 1884 in the
NGC 4526 galaxy
Supernovae Ia are identified through their spectral
signatures. Besides, they show a relatvely simple light
profile

Their spectra and light curves are amazingly uniform,
indicating a common origin and a common intrinsic
luminosity. This is similar to Cepheids – variable stars
which serve as standard candles at relatively short
distances up to about10 Mpc (~30 light years).
1. Measuring the redshift of the spectrum λ /λ0=1+z=1/a one
determines the cosmological redshift z and the scale a at
the time when the observed SN appeared
Cosmological redshift is similar to the Doppler redshift

Doppler redshift
Doppler
redshift

Cosmological
redshift
2. Measuring the apparent magnitude m (logaritamic
measure of the flux) and knowing the absolute magnitude M
(logaritamic measure of the luminosity) one finds the so
called luminosity distance DL

DL  Mpc  10

( m M )/55

1Mpc=3.085 million l.years
Thus for each observed supernova we have two
numbers: the redshift z and the luminosity
distance DL These data are plotted on the
Hubble diagram: z as a function of DL .
Comparing with theoretical curves obtained
from different cosmological models one infers
the history of the Universe expansion.
Hubble diagram
Green line: Empty Model (Ω = 0),
Red : Closed Universe (ΩM = 2 )
Violet : Standard flat model (ΛCDM)
with ΩM = 0.27 and ΩΛ = 0.73
Dashed violet : closed ΛCDM
Black: Enstein-de Sitter Model
(ΩM = 1, ΩΛ = 0, critical universe)
Black dashed: Einstein - de Sitter
Model with Cold DM (p=0)
Blue: de Sitter Model (ΩM = 0,
ΩΛ = 1)
Dashed blue : Evolving Supernovae.
The observations lead to the conclusion that the
Universe expansion is accelerated today. In 2011
the three astrophysicists received the Nobel
prize for physics.
Transition from a decelerated to
accelerated expansion
It is important to look for direct evidence of an earlier, slowing
phase of expansion. Such evidence would help confirm the
standard cosmological model and give scientists a clue to the
underlying cause of the present period of cosmic acceleration.
Because telescopes look back in time as they gather light from
far-off stars and galaxies, astronomers can explore the
expansion history of the universe by focusing on distant objects.
Observing supernovae that appeared 7 billion years ago, i.e., at
the time when the universe was half the present size (a=1/2 or
z=1) shows the Universe expansion was slowing down.
A transition from a decelerated to accelerated expansion
happened at about 4 – 5 billion years ago when the Universe
siye was about a=2/3 (z=0.5) .
Alternative hypotheses
Some skeptical scientists (traditional astronomers in particular)
wondered, whether the teams had correctly interpreted the data
from the supernovae. Was it possible that another effect besides
cosmic acceleration could have caused the supernovae to
appear fainter than expected?
For example, dust filling intergalactic space could also make the
supernovae appear dim. Or perhaps ancient supernovae were
just born dimmer because the chemical composition of the
universe was different from what it is today, with a smaller
abundance of the heavy elements produced by nuclear
reactions in stars.
Test of the alternative hypotheses
Luckily, a good test of the competing hypotheses is available. If
supernovae appear fainter than expected because of an
astrophysical cause, such as a pervasive screen of dust, or
because past supernovae were born dimmer, the dimming
effects should increase with the objects’ redshift. But if the
dimming is the result of a recent cosmic speedup that followed
an earlier era of deceleration, supernovae from the slowdown
period would appear relatively brighter. Therefore, observations
of supernovae that exploded when the universe was less than
two thirds of its present size could provide the evidence to show
which of the hypotheses is correct.
(It is possible, of course, that an unknown astrophysical
phenomenon could precisely match the effects of both the
speedup and slowdown, but scientists generally disfavor such
artificially tuned explanation)
Test of the alternative hypotheses
The Advanced Camera for Surveys, a new imaging instrument
installed on the space telescope in 2002, enabled scientists to
turn Hubble telescop into a supernova-hunting machine.
Riess and his team found 6 supernovae 1a which exploded
when the Universe was less than 1/2 its present size (z > 1,
or more than 7 billion years ago). Together with SN 1997ff,
these are the most distant type Ia supernovae ever
discovered. The observations confirmed the existence of an
early slowdown period and placed the transitional point
between slowdown and speedup at about 5 billion years ago.
This finding is consistent with theoretical expectations of the
standard cosmological model

ΛCDM
Problems with Λ
1) Fine tuning problem. The calculation of the vacuum energy
density in field theory of the Standard Model of particle
physics gives the value about 10120 times higher than the
value of Λ obtained from observations. One possible way out
is fine tuning: a rather unnatural assumption that all
interactions of the standard model of particle physics
somehow conspire to yield cancellation between various
large contributions to the vacuum energy resulting in a small
value of Λ , in agreement with observations
2) Coincidence problem. Why is this fine tuned value of Λ
such that DM and DE are comparable today, leaving one to
rely on anthropic arguments?
9.2 Quintessence
The coincidence problem is somewhat ameliorated in
quintessence – a canonical scalar field with selfinteraction
effectively providing a slow roll inflation today
P.Ratra, J. Peebles PRD 37 (1988)

S   d 4 x L ( X , )



X  g , ,

1
L  X  V ( )
2

2
S
T 
 ,,  L g 

 det g  g

Field theory description of a perfect fluid if X>0

T  (   p)u u  pg 
where

u 

,

1
p  L  X  V ( )
2

X
1
  X  L  X  V ( )
2

A suitable choice of V(φ) yields a desired
cosmology, or vice versa: from a desired equation
of state p=p(ρ) one can derive the Lagrangian of
the corresponding scalar field theory
9.3 Phantom quintessence
Phantom energy is a substance with negative pressure
such that |p| exceeds the energy density so that the null
energy condition (NEC) is violated, i.e., p+ρ<0. Phantom
quintessence is a scalar field with a negative kinetic term

1
p  L   X  V ( )
2

1
   X  V ( )
2

Obviously, for X>0 we have p+ρ<0 which demonstrates a
violation of NEC! This model predicts a catastrophic end of
the Universe, the so-called Big Rip - the total collapse of all
bound systems.
9.4 k-essence
k-essence is a generalized quintessence which was first
introduced as a model for inflation . A minimally coupled
k-essence model is described by
R


4
S =  d x  g 
 L ( , X )
 16G

where L is the most general Lagrangian, which depends
on a single scalar field of dimension , and on the
dimensionless quantity X g  ,, For X>0 , the energy

momentum tensor takes the perfect fluid form

T  = 2LX ,,  Lg  = (   p)u u  p g  ,
where

LX  L/X
The associated hydrodynamic quantities are

p = L( , X )

 = 2 XLX ( , X )  L( , X ).

Examples
Kinetic k-essence The Lagrangian is a function of X
only. In this case

p = L( X )

 = 2 XLX ( X )  L( X )

To this class belong the ghost condensate

L( X ) = A(1  X )2  B
and the scalar Born-Infeld model

L( X ) =  A

1 X
Ghost condensate model
N. Arkani-Hamed et al , JHEP 05 (2004)
R.J. Scherrer, PRL 93 (2004)

L(X)

L(X)
or

X

de-Sitter

X

de-Sitter
Lecture 3
10. DE/DM Unification – Quartessence
The astrophysical and cosmological observational data
can be accommodated by combining baryons with
conventional CDM and a simple cosmological constant
providing DE. This ΛCDM model, however, faces the fine
tuning and coincidence problems associated to Λ.
Another interpretation of this data is that DM/DE are different
manifestations of a common structure. The general class of
models, in which a unification of DM and DE is achieved
through a single entity, is often referred to as quartessence.
Amost of the unification scenarios that have recently been
suggested are basend on k-essence type of models
including Ghost Condensate, various variants of the
Chaplygin Gas and a model called Dusty Dark Energy.
10.1 Chaplygin Gas
An exotic fluid with an equation of state

p

A



The first definite model for a dark matter/energy
unification
A. Kamenshchik, U. Moschella, V. Pasquier, PLB 511 (2001)
N.B., G.B. Tupper, R.D. Viollier, PLB 535 (2002)
J.C. Fabris, S.V.B. Goncalves, P.E. de Souza, GRG 34 (2002)
In a homogeneous model the conservation equation yields the
Chaplygin gas density as a function of the scale factor a

  A B

a6

where B is an integration constant.The Chaplygin gas thus interpolates
between dust (ρ~a -3 ) at large redshifts and a cosmological constant
(ρ~ A½) today and hence yields a correct homogeneous cosmology
Scalar Born-Infeld theory
Chaplygin gas has an equivalent scalar field formulation.
Considering a kinetic k-essence type of Lagrangian

LBI  

A 1 X

X  g  , ,

LBI
T  2
,,  LBI g 
X
perfect fluid

T  (   p)u u  pg 

with

u 

,
X
The Chaplygin gas model is equivalent to (scalar)
Born-Infeld description of a D-brane:
Nambu-Goto action of a p-brane moving in a
p+2 -dimensional bulk

SDBI   A  d p 1 x ( 1) p det( g ind )
the induced metric (“pull back”) of the bulk metric Gab

X a X b
ind
g    Gab 

x x
Xa – coordinates in the bulk, xμ – coordinates on the brane
In 4+1 dim. bulk (p=3) Choose the coordinates
such that
X μ =x μ, μ=0,..3, and let the fifth coordinate
X 4≡ θ be normal to the brane. Then

G  g  for
G 4  0

  0...3
xi

G44  1

x0

Xa

θ
We find a k-essence type of theory

SDBI   A  dx

4

1 X

2



X  g , ,

with



A
1 X 2

p   A 1 X 2

;

and hence

p

A


10.2 Problems with Nonvanishing Sound
Speed
To be able to claim that a field theoretical model actually
achieves unification, one must be assured that initial
perturbations can evolve into a deeply nonlinear regime to
form a gravitational condensate of superparticles that can
play the role of CDM. The inhomogeneous Chaplygin gas
based ona Zel'dovich type approximation has been proposed,
N.B., G.B. Tupper, R.D. Viollier, PLB 535 (2002)

and the picture has emerged that on caustics, where the
density is high, the fluid behaves as cold dark matter, wherea
in voids, w=p/ ρ is driven to the lower bound -1 producing
acceleration as dark energy. In fact, for this issue, the usual
Zel’dovich approximation has the shortcoming that the effects
of finite sound speed are neglected.
In fact, all models that unify DM and DE face the
problem of sound speed related to the well-known
Jeans instability. A fluid with a nonzero sound speed
has a characteristic scale below which the pressure
effectively opposes gravity. Hence the perturbations of
the scale smaller than the sonic horizon will be
prevented from growing.
• Soon after the Chaplygin gas was proposed as a
model of unification it has been shown that the naive
model does not reproduce the mass power spectrum
H.B. Sandvik et al PRD 69 (2004)

and the CMB
D. Carturan and F. Finelli, PRD 68 (2003);
L. Amendola et JCAP 07 (2003)
α<0

α>0

Power spectrum for p=-A/ρα for various α
H.B. Sandvik et al, PRD 2004

α =0
CMB spectrum for p=-A/ρα for various α
L. Amendola et al, JCAP 2003
The physical reason is a nonvanishing sound
speed. Although the adiabatic speed of sound
p
A
2
cs 
 2
 S 
is small until a ~ 1, the accumulated comoving
size of the acoustic horizon
cs

ds   dt  H 0 1a 7 / 2
a
reaches Mpc scales by redshifts of about z ~ 20,
thus frustrating the structure formation at galactic
and subgalactic scales. This may be easily
demonstrated in a simple spherical model.
10.2.1 Spherical Model
To study the evolution of perturbations of a model with
nonvanishing pressure gradients and speed of sound
we will use the spherical model. For a k-essence model
or any one-component type of model, the Euler
equation combines with Einstein’s equations to
 h p, 
3H  3H 2  2  4G(  3 p) = 

 p   ;

2 = 

 = h h u( ;)  H h

We thus obtain an evolution equation for H . Owing to the

definition of the four-velocity and orthogonality h u = 0
we may write




h p, = c h , .
2
s
Hence, if cs =0 or if the pressure gradient is parallel to uμ as
for dust, the right-hand side of the evolution equation and we
obtain

3H  3H 2  2  4G(  3 p) = 0

wihich together with the continuity equation comprises the
original spherical model E. Gaztãnaga and J.A. Lobo, ApJ. 548 (2001).
However, we are not interested in dust, since generally cs ≠0
and hμν p,ν≠0 so we must generalize the spherical model to
include pressure gradients. The density contrast and the
deviation of the Hubble parameter from the background value
are defined by

 = (  ) / 

H = H  H
In the Newtonian approximation subtracting the
background, eliminating δH and ∂tδH and neglecting
shear we find


3 2
4 2
1  
  2H   H (1  ) 


 2
2
31   a xi

 cs2  

=0
 1   xi 

The root of the structure formation problem is the last
term, as may be understood if we solve the evolution
equation at linear order
3 2
cs2


  2 H   H   2 D  0
2
a
with the solution (in k-space)
k – comoving wave number
ds –comoving size of the acoustic
horizon

 k  a 1/ 4 J 5/ 4 (ds k )
behaving asymptotically as

k  a

for

ds k  1

cos (ds k )
k 
for ds k  1
2
a
• The perturbations whose comoving size R=1/k is larger
than ds grow as δ ~ a . Once the perturbations enter the
acoustic horizon, i.e., as soon as R< ds , they undergo
damped oscillations.
In the case of the Chaplygin gas we have

d s  a 7/2/H 0
reaching Mpc scales already at redshifts of order 10-20.
However, small perturbations alone are not the issue, since
large density contrasts are required on galactic and cluster
scales.
Nonlinear evolution of inhomogeneities

As soon as δ ~ 1 the linear perturbation theory cannot be
trusted. An essentially nonperturbative approach is needed in
order to investigate whether a significant fraction of initial
density perturbations collapses in gravitationally bound
structure - the condensate. If that happens the system evolves
into a two-phase structure - a mixture of CDM in the form of
condensate and DE in the form of uncondensed gas.
Two phase structure – mixture of CDM in the form of
condensate and DE in the form of gas
gas + condensate
nearly homogeneous gas
at zdec  1089
collapse during
expansion
To investigate the inhomogeneous Chaplygin gas we
have used the so called Thomas Fermi correspondence:
(to every purely kinetic k-essence there exist a
corresponding canonical complex field theory)
We have derived the solution for the inhomogeneous
Chaplygin-gas cosmology implementing geometric
version of the Zel’dovich approximation: the
transformation from Lagrange to comoving-synchronous
coordinates induces the spatial metric γij
  A

B


γ is the determinant of γij in comoving gauge.
N.B., G.B. Tupper, R.D. Viollier, PLB 535 (2002)
The expectation to achieve a condensate formation based on
the Zel’dovich approximation was naive as it is well known
that Zel’dovich approximation has the shortcoming that the
effects of finite sound speed are neglected.

Later on, using generalizations of the spherical model which
incorporate effects of the acoustic horizon we have shown
that an initially perturbative Chaplygin gas evolves into a
mixed system containing cold dark matter-like gravitational
condensate.
N.B., R. Lindebaum, G.B. Tupper, R.D. Viollier, JCAP 0411 (2003)

3 2
4 2
1  
  2H   H (1  ) 


 2
2
31   a xi

The density contrast
described by a nonlinear
evolution equation either
grows as dust or
undergoes damped
oscillations depending on
the initial conditions in
contrast to the linear
evolution where all
perturbations that enter the
acoustic horizon undergo
dumped oscillations

 cs2  

=0
 1   xi 

Evolution of the density contrast in the spherical
model from aeq = 3 · 10-4 for the comoving
wavelength 1/k = 0.34 Mpc, δk (aeq ) =0.004 (solid)
δk (aeq ) =0.005 (dashed).

N. B., R.J. Lindebaum, G.B. Tupper, and R.D. Viollier, JCAP 11 (2004)
Unfortunately, the estimated collapse fraction of about 1% is
far too small to significantly affect structure formation. This
basically decided the fate of the simple Chaplygin gas model
10.3 Generalized Chaplygin Gas
Another model was proposed in an attempt to solve the
structure formation problem and has gained a wide
popularity. The generalized Chaplygin gas is defined as

p=

A





,

0  1

The additional parameter does afford greater flexibility: e.g.
for small α the sound horizon d s   a 2 /H 0 and thus by fine
tuning α<10-5, the data can be perturbatively accommodated
Other modifications
• The generalized Chaplygin gas in a modified gravity
approach, reminiscent of Cardassian models

8 G
1 1 1 )
H 
A  )
3
2

T. Barreiro and A.A. Sen, PRD 70 (2004)

• A deformation of the Chaplygin gas – Milne-Born-Infeld
theory

L

A X  bX 2

M. Novello, M. Makler, L.S. Werneck and C.A. Romero, PRD 71 (2005)

• Variable Chaplygin gas

p ~  an 

Zong-Kuan Guo, Yuan-Zhong Zhang, astro-ph/0506091, PLB (2007)
10.4 Tachyon Condensate
The failure of the simple Chaplygin gas (CG) does not exhaust
all the possibilities for quartessence. The Born-Infeld
Lagrangian is a special case of the string-theory inspired
tachyon Lagrangian in which the constant A is replaced by a
potential

L = V ()

1  g  ,, .

Tachyon models are a particular case of k-essence. It was
noted in Zong-Kuan Guo, et al astro-ph/0506091, PLB (2007) that (in the
FRW cosmology), the tachyon model is described by the CG
equation of state in which the constant A is replaced by a
n
function of the cosmological scale factor p ~  a  so the
model was dubbed “variable Chaplygin gas”.
A preliminary analysis of a unifying model based on the
tachyon type Lagrangian has been carried out in
N.B., G.B. Tupper, R.D. Viollier, PRD 80 (2009)

for a potential of the form

V ( )  Vn 2 n

n  0,1,2

n=0 gives the Dirac-Born-Infeld description of a D-brane
- equivalent to the Chaplygin gas
It may be shown that the model with n≠0 effectively behaves
as a variable Chaplygin gas, with p ~  a 6n  . The much
smaller sonic horizon d s ~ a (7 23n ) H 0 enhances condensate
formation by 2 orders of magnitude over the simple Chaplygin
gas. Hence this type of model may salvage the quartessence
scenario.
10.5 Entropy Perturbations
One way to deal with the structure formation problem, is to
assume entropy perturbations such that the effective speed of
sound cs vanishes. In that picture we assume that A is not a
constant so we have
p = cs2  A/ = 0
even if cs≠0. But in a single field model it is precisely the
adiabatic speed of sound that governs the evolution. Hence,
entropy perturbations require the introduction of a second field
on which A depends.
10.5 Entropy Perturbations
One way to deal with the structure formation problem, is to
assume entropy perturbations such that the effective speed
of sound cs vanishes. This scenario has been suggested as
a possible way out of the structure formation problem
immanent to all DM/DE unification models. It has been
noted by R. Reis et al Phys. Rev. D 69 (2004) that the root of the
structure formation problem is the term ∇2δp in perturbation
equations, equal to cs2∇2δ for adiabatic perturbations, and if
there are entropy perturbations such that δp = 0, no
difficulty arises.
Suppose that the matter
Lagrangian depends on two
degrees of freedom, e.g., a
Born-Infeld scalar field θ
and one additional scalar
field φ. In this case, instead
of a simple barotropic form
p=p(ρ), the equation of
state is parametric
p=p(θ,φ) ρ=ρ(θ,φ)
and involves the entropy
density (entropy per
particle) s=s(θ,φ)

The corresponding
perturbations

p
p
 p  
 


s
s
 s  
 


The speed of sound is the sum of two nonadiabatic
terms
1

cs2 

p
p s  s  p


  S 0      
 

Thus, even for a nonzero ∂p/∂ the speed of sound
may vanish if the second term on the right-hand side
cancels the first one. This cancellation will take place if
in the course of an adiabatic expansion, the
perturbation δφ grows with a in the same way as δ .
In this case, it is only a matter of adjusting the initial
conditions of δφ with δ to get cs=0.
Obviously, this scenario cannot work for simple or
generalized Chaplygin gas models as these models are
adiabatic.
We have made attempts at realizing the nonadiabatic
scenario in some 2 component models (hybrid Chaplygin
gas and Kalb- Ramond Chaplygin gas)
N. B., G.B. Tupper, and R.D. Viollier, JCAP 0510 (2005);
N. B., G.B. Tupper, and R.D. Viollier, J. Phys. A40,(2007),

Aside from negating the simplicity of the one-field model,
our analysis has convinced us that even if δp=0 is arranged
as an initial condition, it is all but impossible to maintain this
condition in a realistic model for evolution.
10.6 Dusty Dark Energy
Another way to bypass the structure formation problem is to
impose a constraint on pressure such that the pressure
gradient is parallel to the fluid 4-velocity. The model called
Dusty Dark Energy comprises two scalar fields λ and φ,
λ being a Lagrange multiplier which enforces a constraint
between φ and its kinetic energy term X . Starting from the
action

1

S =  d 4 x  g  K  , X )    X  V  ),
2



where K (φ,X) is an arbitrary function of X and φ. The field
λ is a Lagrange multiplier and does not have a kinetic
term, while X is a standard kinetic term for φ.
The energy momentum tensor takes the usual perfect fluid
form with the 4-velocity as before u = , / X
The λ field equation

1 S 1
= X  V ( ) = 0
 g  2
imposes a constraint such that the pressure

p = K ( , X ) = K ( ,2V ( ))
becomes a function of φ only. Then, its gradient is proportional
to φ,μ and hence parallel to the 4-velocity. In this way cs is
always identically zero. In particular, cosmological
perturbations reproduce the standard hydrodynamic behavior
in the limit of vanishing cs . In a certain limit this model
reproduces the evolution history of ΛCDM, with some
potentially measurable differences.
Fate of the universe
Solving the mystery of cosmic acceleration will reveal the
destiny of our universe. Possible scenarios:
1. If the dark energy density ρΛ is constant or increasing with
time, in 100 billion years or so all but a few hundred
galaxies will be far too redshifted to be seen.
2. But if the dark energy density decreases and matter
becomes dominant again, our cosmic horizon will grow,
revealing more of the universe.
3. Even more extreme (and lethal) futures are possible. If
dark energy density increases rather than decreases, the
universe will eventually undergo a “hyper speedup” that
would result in a Big Rip: tearing apart all bound systems,
galaxies, solar systems, planets and atomic nuclei, in that
order.
11 Conclusions and Outlook
 We have discussed a just a few out of many attempts to
unify DE/DM. The above mentioned examples illustrate
more or less conventional trends in modern cosmology.
 There exist various alternative ideas such as modified
theories of gravity. One of the popular ideas is the socalled brane world cosmology where our world is a fourdimensional membrane submerged in a five-dimensional
space.
 Alternative theories explain more or less successfully a
part of the phenomena related to dark energy and dark
matter, but up until now there is no completely satisfactory
theory which would solve all the puzzles.
 In any case, there remains a lot of work to be done for
theoretical physicists.
Neven Bilic, "Dark Matter, Dark Energy, and Unification Models"
Thomas-Fermi correspondence
Under reasonable assumptions in the cosmological
context there exist an equivalence

Complex scalar field
theories (canonical or
phantom)

Kinetic k-essence type
of models
Consider

L g




 ,  V (|  | m )
*
,

2

2

Thomas-Fermi approximation




2

e

 im

,  m

,  i,

TF Lagrangian
where

X g  ,,


LTF  4  XY  U (Y )
/m
Y

2
2m

2

U (Y )  V (Y ) / m4
Equations of motion for φ and θ

U
X
0
Y



(Yg , );  0

We now define the potential W(X) through a Legendre
transformation

W ( X )  U (Y )  XY
with

X  UY

and

Y  WX

U
UY 
Y
W
WX 
X
correspondence
Complex scalar FT

L g  *,,  V (|  |2




2

Kinetic k-essence FT
m2 )

e im

Eqs. of motion
( 2 g  , );  0
1 dV
g ,,  2
0
2
m d | |


L  m4W ( X )
X  g  , ,

Eq. of motion
(WX g  , );  0
Parametric eq. of state
p  m4W

  m4 (2 XWX  W )
Current conservation
Klein-Gordon current
  ig  (*,  * )
j
,

kinetic k-essence current
j   2m2WX g  ,

U(1) symmetry

shift symmetry

  ei 

   c
Example: Quartic potential
Scalar field potential
2
V  0  m0 |  |2  |  |4
V

2

U (Y ) 

1
1 
1
Y
 2


2
2  8

Kinetic k-essence
1
1 
W(X )   X 

2
2 

2
Example: Chaplygin gas
Scalar field potential

 |  |2
m2 
4
V m  2 

2 
| | 
 m

Scalar Born-Infeld FT

1

U (Y )   Y  
Y


W ( X )  2 1  X

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Neven Bilic, "Dark Matter, Dark Energy, and Unification Models"

  • 1. Dark Matter, Dark Energy, and Unification Models Neven Bilić Ruđer Bošković Institute Zagreb
  • 2. Lecture 1 1 2 3 4 5 BIG BANG History of the Early Universe - Universe Creation History of the Early Universe – Inflation Cosmological Observations and History of the Universe Empirical Grounds for Observational Cosmology 5.1 Hubble’s Law 5.2 Cosmic Microwave Background (CMB) Radiation 5.3 Big Bang Nucleosynthesis 6 Theoretical Foundation of Modern Cosmology 7 Density of Matter in Space 8 Dark Matter
  • 3. Lecture 2 9 Dark Energy 9.1 Vacuum Energy – Cosmological Constant 9.2 Quintessence 9.3 Phantom quintessence 9.4 k-essence
  • 4. Lecture 3 10 DE/DM Unification – Quartessence 10.1 Chaplygin gas 10.2 Problems with Nonvanishing Sound Speed 10.2.1 Spherical Model 10.3 Generalized Chaplygin Gas 10.4 Tachyon Condensate 10.5 Entropy Perturbations 10.6 Dusty Dark Energy 11 Conclusions and Outlook
  • 5. 1. Big Bang What kind of explosion is the Big Bang? The Big Bang is like an explosion of a bomb that took place in a previously empty space. Wrong! The Big Bang is an eplosion of space itself which, before that, was concentrated in one point. Right! C.H. Lineweaver and T. Davis, Misconceptions About the BIG BANG, Scientific American, March 2005
  • 6. 2. Early Universe - Creation The Universe begins as a quantum fluctuation of the initial vacuum The total energy of the Universe (matter energy+dark energy +gravity) = 0 Owing to the uncertainty relations DE Dt  h h – Planck constant Borrowing a small amount of energy (E0 ) from the vacuum is allowed for a long time (t  )! As a consequence, the lifetime of the Universe is almost infinite
  • 7. 3. Early Universe - Inflation A short period of inflation follows – very rapid expansion 1025 times in 10-32 s.
  • 9. The horizon problem. Observations of the cosmic microwave background radiation (CMB) show that the Universe is homogeneous and isotropic. The problem arises because the information about CMB radiation arrive from distant regions of the Universe which were not in a causal contact at the moment when radiation had been emitted – in contradiction with the observational fact that the measured temperature of radiation is equal (up to the deviations of at most about 10-5) in all directions of observation.
  • 10. The flatness problem Observations of the average matter density, expansion rate and fluctuations of the CMB radiation show that the Universe is flat or with a very small curvature today. In order to achieve this, a “fine-tuning“ of the initial conditions is needed, which is rather unnatural. The answer is given by inflation:
  • 11.  v =5 105Hd , The initial density perturbations problem. The question is how the initial deviations from homogeneity of the density are formed having in mind that they should be about 10-5 in order to yield today’s structures (stars, galaxies, clusters). The answer is given by inflation: perturbations of density are created as quantum fluctuations of the inflaton field.
  • 12. 4. Cosmological Observations and History of the Universe • • • • • • The properties we would like to find out: Spacetime geometry Age of the Universe Expansion rate Amount of ordinary (“baryonic”) matter Amount and nature of dark matter and dark energy Mechanism of structure formation • • • • • • Observations that could yield these properties: Background microwave radiation Observations of distant dalekih supernovae Large scale structure observations Concentrations of light elements Galactic rotation curves Gravitational lenses
  • 13. 13 billion years Universe Today 5 billion yeras Solar System Foramation, First Supernove 1 billion years Galaxy and Star Formation 100 000 years 100 s 10-10 s 10-34 s Formation of Atoms, Decupling of Matter–Radiation Nucleosynthesis of Light Elements Synthesis of Protons and Neutrons QCD Plasma: quarks and gluons Electroweak Transition End of Inflation 10-43 s Quantum Gravity Era
  • 14. After a short period of inflation the Universe continue expanding at much slower rate so that after roughly 13.5 billion years, the visible part is about 90 billion light years.
  • 15. 5. Empirical Grounds for Observational Cosmology • The expansion of the Universe - Hubble’s law v = Hd • Cosmic microwave background radiation very homogeneous in all directions • Big Bang nucleosynthesis - proportions of light elements (H, D, He, Li)
  • 17. Hubble’s law – Universe expansion a=0.5
  • 18. Hubble’s law – Universe expansion a=0.75
  • 20. Hubble’s law – Universe expansion recession velocity a=0.5 a=0.75 a=1
  • 21. CAN GALAXIES RECEDE FASTER THAN LIGHT? Of course not. Einstein’s special theory of relativity forbids that. Wrong! Sure they can. Special relativity does not apply to recession velocity. Right! RH RH - Hubble radius
  • 22. CAN WE SEE GALAXIES RECEDING FASTER THAN LIGHT? Of course not. Light from those galaxies never reaches us. Wrong! Sure we can, because the expansion rate changes over time. Right!
  • 23. Do objects inside the Universe expand, too? Yes. Expansion causes the universe and everything in it to grow. Wrong! Ne. No. The universe grows, but coherent objects (stars, galaxies ..) do not Right!
  • 24. T = 2.723K DT = 100K DT = 200K Measuring CMB; the temperature map of the sky.
  • 25. Angular (multipole) spectrum of the fluctuations of the CMB (Planck 2013)
  • 26. 6. Theoretical Foundation of Modern Cosmology • General relativity – gravity G=T Matter determines the space-time geometry Geometry determines the motion of matter • Homogeneity and isotropy of space – approximate property on very large scales (~Glyrs today) • Fluctuations of matter and geometry in the early Universe cause structure formation (stars, galaxys, clusters ...)
  • 27. General Relativity – Gravity 1 R  g  R  8 T  g   2 g - metric tensor R - Riemann curvature tensor  - cosmological constant T  - energy – momentum tensor
  • 28. Perfect Fluid Matter is described by a perfect fluid T  (   p)u u  pg  u - fluid velocity T- energy-momentum tensor p - pressure ρ - energy density
  • 29. The energy momentum conservation T  ; = 0 yields, as its longitudinal part uT  ; = 0 , the continuity equation    3H (   p) = 0, and, as its transverse part, the Euler equation  u = 1 h   p, , p where 3H = u ; ; h  = g   u u   = u , ;  u  = u u  ; . projector onto the three-space orthogonal to H is the local Hubble parameter. u
  • 30. Cosmological principle Homogeneity and isotropy of space  dr 2 2 2 ds  dt  a (t )   r d  2 1  kr  2 2 2 - cosmological scale the curvature constant k takes on the values 1, 0, or -1, for a closed, flat, or open universe, respectively. a (t ) – cosmological scale
  • 31. Friedman equations  k 8 G a  2     3 a a 4 G    a (   3 p) 3 2 T  ;  0  a H a k expansion rate   0 closed  0 flat  0 open   a 3(1 w) w p  Various kinds of the cosmic fluids with different w • radiation • matter • vacuum pR   R 3 pM  0 p     w 1 3 w0 w  1   a 4   a 3   a0
  • 32. 7. Density of Matter in Space The best agreement with cosmologic observations are obtained by the models with a flat space According to Einstein’s theory, a flat space universe requires critical matter density cr today cr  10-29 g/cm3 Ω=  / cr ratio of the actual to the critical density For a flat space Ω=1
  • 33. What does the Universe consist of? From astronomical observations: luminous matter (stars, galaxies, gas ...) lum/cr0.5% From the light element abundances and comparison with the Big Bang nucleosynthesis: barynic matter(protons, neutrons, nuclei) Bar/cr5% Total matter density fraction ΩM=M/cr 0.31 Accelerated expansion and comparison of the standard Big Bang model with observations requires that the dark energy density (vacuum energy) today ΩΛ=  /cr = 0. 69%
  • 34. Density fractions of various kinds of matter today with respect to the total density B B   0.05  tot DM  DM   0.26  tot     0.69  tot These fractions change with time but for a spatialy flat Universe the following always holds:  tot  crit B  DM    1
  • 35. Age of the Universe Easy to calculate using the present observed fractions of matter, radiation and vacuum energy. For a spatially flat Universe from the Friedmann equation and energy consrvation we have H (a)  H 0 (  M a 3  Ra 4 )1/2 H 0  h  100Gpc/s2  (14.5942Gyr)1, h  0.67 Age of the Universe is then T 1 1 da 1 da a T   dt     (a 3  M )1/2 aH H 0 0 0 0 T  13.78Gyr
  • 36. How Large is the Observable Universe? S The universe is 14 billion years old, so the radius of the observable part is 14 billion lightyears. Wrong! Because space is expanding, the observable part of our Universe has a radius of more than 14 billion light-years. Right!
  • 37. 8. Dark nature of the Universe – Dark Matter • • • • According to present recent obsrvations (Planck Satellite Mission): More than 99% of matter is not luminous Out of that less then 5% is ordinary (“baryonic”) About 26% is Dark Matter About 61% is Dark Energy (Vacuum Energy) 0.69 0.26 0.04 0.01
  • 39. Cluster 1E0657-558, Bullet Clowe et al, astro-ph/0608407 http://chandra.harvard.edu
  • 42. DM contents - possible candidates Baryonic DM in the form of astrophysical objects • • • • Brown dwarfs Black holes MACHO planets Nonbaryonic DM • Sterile neutrino • Axion • SUSY stable particles: gravitino, neutralino, axino Exsperimental evidence? - great expectations from the LHC experiments at CERN
  • 43. Hot DM refers to low-mass neutral particles that are still relativistic when galaxy-size masses (  1012 M  ) are first encompassed within the horizon. Hence, fluctuations on galaxy scales are wiped out. Standard examples of hot DM are neutrinos and majorons. They are still in thermal equilibrium after the QCD deconfinement transition, which took place at TQCD ≈ 150 MeV. Hot DM particles have a cosmological number density comparable with that of microwave background photons, which implies an upper bound to their mass of a few tens of eV.
  • 44. Warm DM particles are just becoming nonrelativistic when galaxy-size masses enter the horizon. Warm DM particles interact much more weakly than neutrinos. They decouple (i.e., their mean free path first exceeds the horizon size) at T>>TQCD. As a consequence, their mass is expected to be roughly an order of magnitude larger, than hot DM particles. Examples of warm DM are keV sterile neutrino, axino, or gravitino in soft supersymmetry breaking scenarios.
  • 45. Cold DM particles are already nonrelativistic 6 when even globular cluster masses (  10 M ) enter the horizon. Hence, their free is of no cosmological importance. In other words, all cosmologically relevant fluctuations survive in a universe dominated by cold DM. The two main particle candidates for cold dark matter are the lowest supersymmetric weakly interacting massive particles (WIMPs) and the axion.
  • 46. 9. Dark Energy Because gravity acts as an attractive force between astrophysical objects we expect that the expansion of the Universe will slowly decelerate. However, recent observations indicate that the Universe expansion began to accelerate since about 5 billion years ago. Repulsive gravity?
  • 48. Accelerated expansion   0 One possible explanation is the existence of a fluid with negative pressure such that p  3  0 and in the second Friedmann equation the universe  acceleration a becomes positive cosmological constant   vacuum energy density with equation of state p=-ρ. Its negative pressure may be responsible for accelerated expansion! New term: Dark Energy – fluid with negative presssure - generalization of the concept of vacuum energy
  • 50. Time dependence of the DE density Another important property of DE is that its density does not vary with time or very weakly depends on time. In contrast , the density of ordinary matter varies rapidly because of a rapid volume expansion. The raughh picture is that in the early Universe when the density of matter exceeded the density of DE the Universe expansion was slowing down. In the course of evolution the matter density decreases and when the DE density began to dominate, the Universe began to accelerate.
  • 51. Most popular models of dark energy • Cosmological constant – vacuum energy density. Energy density does not vary with time. • Quintessence – a scalar field with a canonical kinetic term. Energy density varies with time. • Phantom quintessence – a scalar field with a negative kinetic term. Energy density varies with time. • k-essence – a scalar field whose Lagrangian is a general function of kinetic energy. Energy density varies with time. • Quartessence – a model of unifying of DE and DM. Special subclass of k-essence. One of the popular models is the so-called Chaplygin gas
  • 52. 9.1 Cosmological Constant – Einstein’s biggest mistake? 1917 Einstein proposes cosmological constant 1929 Hubble discovers expansion of the Universe 1934 Einstein declares the cosmological constant his biggest blunder 1998 Pearlmutter, Riess, and Schmidt find evidence for the cosmological constant or dark energy
  • 53. Origin of the cosmological constant: Damn! The vacuum is not empty. Quantum fluctuations of the vacuum W+ W- e+ quark eantiquark particle antiparticle
  • 54. Observations of supernovae type Ia Od 1988 Saul Perlmutter (Univ. Of California) lieder of Supernova Cosmology Project Od 1994, Brian Schmidt (Australian National University) klieder of High z Supernova Search Team In the same team Adam Riess (Space Telescope Science Institute) the first one who made the analysis
  • 55. Supernovae Ia occure in binary systems made of a white dwarf and a large companion star. A low-mass white dwarf accreting matter from a nearby companion approaches the limit of 1.4 solar masses (Chandrasekar limit) and becomes unstable. Then, a thermonuclear explosion takes place ensues and an immense amount of energy is suddenly released.
  • 56. In a typical galaxy, supernovae occur a few times in thousand years. In a short period a huge amount of energy is released After a few days the emission achieves a maximum with brightness comparable with the luminosity of an entire galaxy. Supernova SN 1884 in the NGC 4526 galaxy
  • 57. Supernovae Ia are identified through their spectral signatures. Besides, they show a relatvely simple light profile Their spectra and light curves are amazingly uniform, indicating a common origin and a common intrinsic luminosity. This is similar to Cepheids – variable stars which serve as standard candles at relatively short distances up to about10 Mpc (~30 light years).
  • 58. 1. Measuring the redshift of the spectrum λ /λ0=1+z=1/a one determines the cosmological redshift z and the scale a at the time when the observed SN appeared Cosmological redshift is similar to the Doppler redshift Doppler redshift
  • 60. 2. Measuring the apparent magnitude m (logaritamic measure of the flux) and knowing the absolute magnitude M (logaritamic measure of the luminosity) one finds the so called luminosity distance DL DL  Mpc  10 ( m M )/55 1Mpc=3.085 million l.years
  • 61. Thus for each observed supernova we have two numbers: the redshift z and the luminosity distance DL These data are plotted on the Hubble diagram: z as a function of DL . Comparing with theoretical curves obtained from different cosmological models one infers the history of the Universe expansion.
  • 62. Hubble diagram Green line: Empty Model (Ω = 0), Red : Closed Universe (ΩM = 2 ) Violet : Standard flat model (ΛCDM) with ΩM = 0.27 and ΩΛ = 0.73 Dashed violet : closed ΛCDM Black: Enstein-de Sitter Model (ΩM = 1, ΩΛ = 0, critical universe) Black dashed: Einstein - de Sitter Model with Cold DM (p=0) Blue: de Sitter Model (ΩM = 0, ΩΛ = 1) Dashed blue : Evolving Supernovae.
  • 63. The observations lead to the conclusion that the Universe expansion is accelerated today. In 2011 the three astrophysicists received the Nobel prize for physics.
  • 64. Transition from a decelerated to accelerated expansion It is important to look for direct evidence of an earlier, slowing phase of expansion. Such evidence would help confirm the standard cosmological model and give scientists a clue to the underlying cause of the present period of cosmic acceleration. Because telescopes look back in time as they gather light from far-off stars and galaxies, astronomers can explore the expansion history of the universe by focusing on distant objects. Observing supernovae that appeared 7 billion years ago, i.e., at the time when the universe was half the present size (a=1/2 or z=1) shows the Universe expansion was slowing down. A transition from a decelerated to accelerated expansion happened at about 4 – 5 billion years ago when the Universe siye was about a=2/3 (z=0.5) .
  • 65. Alternative hypotheses Some skeptical scientists (traditional astronomers in particular) wondered, whether the teams had correctly interpreted the data from the supernovae. Was it possible that another effect besides cosmic acceleration could have caused the supernovae to appear fainter than expected? For example, dust filling intergalactic space could also make the supernovae appear dim. Or perhaps ancient supernovae were just born dimmer because the chemical composition of the universe was different from what it is today, with a smaller abundance of the heavy elements produced by nuclear reactions in stars.
  • 66. Test of the alternative hypotheses Luckily, a good test of the competing hypotheses is available. If supernovae appear fainter than expected because of an astrophysical cause, such as a pervasive screen of dust, or because past supernovae were born dimmer, the dimming effects should increase with the objects’ redshift. But if the dimming is the result of a recent cosmic speedup that followed an earlier era of deceleration, supernovae from the slowdown period would appear relatively brighter. Therefore, observations of supernovae that exploded when the universe was less than two thirds of its present size could provide the evidence to show which of the hypotheses is correct. (It is possible, of course, that an unknown astrophysical phenomenon could precisely match the effects of both the speedup and slowdown, but scientists generally disfavor such artificially tuned explanation)
  • 67. Test of the alternative hypotheses The Advanced Camera for Surveys, a new imaging instrument installed on the space telescope in 2002, enabled scientists to turn Hubble telescop into a supernova-hunting machine. Riess and his team found 6 supernovae 1a which exploded when the Universe was less than 1/2 its present size (z > 1, or more than 7 billion years ago). Together with SN 1997ff, these are the most distant type Ia supernovae ever discovered. The observations confirmed the existence of an early slowdown period and placed the transitional point between slowdown and speedup at about 5 billion years ago. This finding is consistent with theoretical expectations of the standard cosmological model ΛCDM
  • 68. Problems with Λ 1) Fine tuning problem. The calculation of the vacuum energy density in field theory of the Standard Model of particle physics gives the value about 10120 times higher than the value of Λ obtained from observations. One possible way out is fine tuning: a rather unnatural assumption that all interactions of the standard model of particle physics somehow conspire to yield cancellation between various large contributions to the vacuum energy resulting in a small value of Λ , in agreement with observations 2) Coincidence problem. Why is this fine tuned value of Λ such that DM and DE are comparable today, leaving one to rely on anthropic arguments?
  • 69. 9.2 Quintessence The coincidence problem is somewhat ameliorated in quintessence – a canonical scalar field with selfinteraction effectively providing a slow roll inflation today P.Ratra, J. Peebles PRD 37 (1988) S   d 4 x L ( X , )  X  g , , 1 L  X  V ( ) 2 2 S T   ,,  L g    det g  g Field theory description of a perfect fluid if X>0 T  (   p)u u  pg 
  • 70. where u  , 1 p  L  X  V ( ) 2 X 1   X  L  X  V ( ) 2 A suitable choice of V(φ) yields a desired cosmology, or vice versa: from a desired equation of state p=p(ρ) one can derive the Lagrangian of the corresponding scalar field theory
  • 71. 9.3 Phantom quintessence Phantom energy is a substance with negative pressure such that |p| exceeds the energy density so that the null energy condition (NEC) is violated, i.e., p+ρ<0. Phantom quintessence is a scalar field with a negative kinetic term 1 p  L   X  V ( ) 2 1    X  V ( ) 2 Obviously, for X>0 we have p+ρ<0 which demonstrates a violation of NEC! This model predicts a catastrophic end of the Universe, the so-called Big Rip - the total collapse of all bound systems.
  • 72. 9.4 k-essence k-essence is a generalized quintessence which was first introduced as a model for inflation . A minimally coupled k-essence model is described by R   4 S =  d x  g   L ( , X )  16G  where L is the most general Lagrangian, which depends on a single scalar field of dimension , and on the dimensionless quantity X g  ,, For X>0 , the energy  momentum tensor takes the perfect fluid form T  = 2LX ,,  Lg  = (   p)u u  p g  , where LX  L/X
  • 73. The associated hydrodynamic quantities are p = L( , X )  = 2 XLX ( , X )  L( , X ). Examples Kinetic k-essence The Lagrangian is a function of X only. In this case p = L( X )  = 2 XLX ( X )  L( X ) To this class belong the ghost condensate L( X ) = A(1  X )2  B and the scalar Born-Infeld model L( X ) =  A 1 X
  • 74. Ghost condensate model N. Arkani-Hamed et al , JHEP 05 (2004) R.J. Scherrer, PRL 93 (2004) L(X) L(X) or X de-Sitter X de-Sitter
  • 76. 10. DE/DM Unification – Quartessence The astrophysical and cosmological observational data can be accommodated by combining baryons with conventional CDM and a simple cosmological constant providing DE. This ΛCDM model, however, faces the fine tuning and coincidence problems associated to Λ. Another interpretation of this data is that DM/DE are different manifestations of a common structure. The general class of models, in which a unification of DM and DE is achieved through a single entity, is often referred to as quartessence. Amost of the unification scenarios that have recently been suggested are basend on k-essence type of models including Ghost Condensate, various variants of the Chaplygin Gas and a model called Dusty Dark Energy.
  • 77. 10.1 Chaplygin Gas An exotic fluid with an equation of state p A  The first definite model for a dark matter/energy unification A. Kamenshchik, U. Moschella, V. Pasquier, PLB 511 (2001) N.B., G.B. Tupper, R.D. Viollier, PLB 535 (2002) J.C. Fabris, S.V.B. Goncalves, P.E. de Souza, GRG 34 (2002)
  • 78. In a homogeneous model the conservation equation yields the Chaplygin gas density as a function of the scale factor a   A B a6 where B is an integration constant.The Chaplygin gas thus interpolates between dust (ρ~a -3 ) at large redshifts and a cosmological constant (ρ~ A½) today and hence yields a correct homogeneous cosmology
  • 79. Scalar Born-Infeld theory Chaplygin gas has an equivalent scalar field formulation. Considering a kinetic k-essence type of Lagrangian LBI   A 1 X X  g  , , LBI T  2 ,,  LBI g  X perfect fluid T  (   p)u u  pg  with u  , X
  • 80. The Chaplygin gas model is equivalent to (scalar) Born-Infeld description of a D-brane: Nambu-Goto action of a p-brane moving in a p+2 -dimensional bulk SDBI   A  d p 1 x ( 1) p det( g ind ) the induced metric (“pull back”) of the bulk metric Gab X a X b ind g    Gab   x x Xa – coordinates in the bulk, xμ – coordinates on the brane
  • 81. In 4+1 dim. bulk (p=3) Choose the coordinates such that X μ =x μ, μ=0,..3, and let the fifth coordinate X 4≡ θ be normal to the brane. Then G  g  for G 4  0   0...3 xi G44  1 x0 Xa θ
  • 82. We find a k-essence type of theory SDBI   A  dx 4 1 X 2  X  g , , with  A 1 X 2 p   A 1 X 2 ; and hence p A 
  • 83. 10.2 Problems with Nonvanishing Sound Speed To be able to claim that a field theoretical model actually achieves unification, one must be assured that initial perturbations can evolve into a deeply nonlinear regime to form a gravitational condensate of superparticles that can play the role of CDM. The inhomogeneous Chaplygin gas based ona Zel'dovich type approximation has been proposed, N.B., G.B. Tupper, R.D. Viollier, PLB 535 (2002) and the picture has emerged that on caustics, where the density is high, the fluid behaves as cold dark matter, wherea in voids, w=p/ ρ is driven to the lower bound -1 producing acceleration as dark energy. In fact, for this issue, the usual Zel’dovich approximation has the shortcoming that the effects of finite sound speed are neglected.
  • 84. In fact, all models that unify DM and DE face the problem of sound speed related to the well-known Jeans instability. A fluid with a nonzero sound speed has a characteristic scale below which the pressure effectively opposes gravity. Hence the perturbations of the scale smaller than the sonic horizon will be prevented from growing. • Soon after the Chaplygin gas was proposed as a model of unification it has been shown that the naive model does not reproduce the mass power spectrum H.B. Sandvik et al PRD 69 (2004) and the CMB D. Carturan and F. Finelli, PRD 68 (2003); L. Amendola et JCAP 07 (2003)
  • 85. α<0 α>0 Power spectrum for p=-A/ρα for various α H.B. Sandvik et al, PRD 2004 α =0
  • 86. CMB spectrum for p=-A/ρα for various α L. Amendola et al, JCAP 2003
  • 87. The physical reason is a nonvanishing sound speed. Although the adiabatic speed of sound p A 2 cs   2  S  is small until a ~ 1, the accumulated comoving size of the acoustic horizon cs  ds   dt  H 0 1a 7 / 2 a reaches Mpc scales by redshifts of about z ~ 20, thus frustrating the structure formation at galactic and subgalactic scales. This may be easily demonstrated in a simple spherical model.
  • 88. 10.2.1 Spherical Model To study the evolution of perturbations of a model with nonvanishing pressure gradients and speed of sound we will use the spherical model. For a k-essence model or any one-component type of model, the Euler equation combines with Einstein’s equations to  h p,  3H  3H 2  2  4G(  3 p) =    p   ; 2 =   = h h u( ;)  H h We thus obtain an evolution equation for H . Owing to the  definition of the four-velocity and orthogonality h u = 0 we may write   h p, = c h , . 2 s
  • 89. Hence, if cs =0 or if the pressure gradient is parallel to uμ as for dust, the right-hand side of the evolution equation and we obtain 3H  3H 2  2  4G(  3 p) = 0 wihich together with the continuity equation comprises the original spherical model E. Gaztãnaga and J.A. Lobo, ApJ. 548 (2001). However, we are not interested in dust, since generally cs ≠0 and hμν p,ν≠0 so we must generalize the spherical model to include pressure gradients. The density contrast and the deviation of the Hubble parameter from the background value are defined by  = (  ) /  H = H  H
  • 90. In the Newtonian approximation subtracting the background, eliminating δH and ∂tδH and neglecting shear we find  3 2 4 2 1     2H   H (1  )     2 2 31   a xi  cs2    =0  1   xi  The root of the structure formation problem is the last term, as may be understood if we solve the evolution equation at linear order
  • 91. 3 2 cs2     2 H   H   2 D  0 2 a with the solution (in k-space) k – comoving wave number ds –comoving size of the acoustic horizon  k  a 1/ 4 J 5/ 4 (ds k ) behaving asymptotically as k  a for ds k  1 cos (ds k ) k  for ds k  1 2 a • The perturbations whose comoving size R=1/k is larger than ds grow as δ ~ a . Once the perturbations enter the acoustic horizon, i.e., as soon as R< ds , they undergo damped oscillations.
  • 92. In the case of the Chaplygin gas we have d s  a 7/2/H 0 reaching Mpc scales already at redshifts of order 10-20. However, small perturbations alone are not the issue, since large density contrasts are required on galactic and cluster scales.
  • 93. Nonlinear evolution of inhomogeneities As soon as δ ~ 1 the linear perturbation theory cannot be trusted. An essentially nonperturbative approach is needed in order to investigate whether a significant fraction of initial density perturbations collapses in gravitationally bound structure - the condensate. If that happens the system evolves into a two-phase structure - a mixture of CDM in the form of condensate and DE in the form of uncondensed gas.
  • 94. Two phase structure – mixture of CDM in the form of condensate and DE in the form of gas gas + condensate nearly homogeneous gas at zdec  1089 collapse during expansion
  • 95. To investigate the inhomogeneous Chaplygin gas we have used the so called Thomas Fermi correspondence: (to every purely kinetic k-essence there exist a corresponding canonical complex field theory) We have derived the solution for the inhomogeneous Chaplygin-gas cosmology implementing geometric version of the Zel’dovich approximation: the transformation from Lagrange to comoving-synchronous coordinates induces the spatial metric γij   A B  γ is the determinant of γij in comoving gauge. N.B., G.B. Tupper, R.D. Viollier, PLB 535 (2002)
  • 96. The expectation to achieve a condensate formation based on the Zel’dovich approximation was naive as it is well known that Zel’dovich approximation has the shortcoming that the effects of finite sound speed are neglected. Later on, using generalizations of the spherical model which incorporate effects of the acoustic horizon we have shown that an initially perturbative Chaplygin gas evolves into a mixed system containing cold dark matter-like gravitational condensate. N.B., R. Lindebaum, G.B. Tupper, R.D. Viollier, JCAP 0411 (2003)
  • 97.  3 2 4 2 1     2H   H (1  )     2 2 31   a xi The density contrast described by a nonlinear evolution equation either grows as dust or undergoes damped oscillations depending on the initial conditions in contrast to the linear evolution where all perturbations that enter the acoustic horizon undergo dumped oscillations  cs2    =0  1   xi  Evolution of the density contrast in the spherical model from aeq = 3 · 10-4 for the comoving wavelength 1/k = 0.34 Mpc, δk (aeq ) =0.004 (solid) δk (aeq ) =0.005 (dashed). N. B., R.J. Lindebaum, G.B. Tupper, and R.D. Viollier, JCAP 11 (2004)
  • 98. Unfortunately, the estimated collapse fraction of about 1% is far too small to significantly affect structure formation. This basically decided the fate of the simple Chaplygin gas model
  • 99. 10.3 Generalized Chaplygin Gas Another model was proposed in an attempt to solve the structure formation problem and has gained a wide popularity. The generalized Chaplygin gas is defined as p= A   , 0  1 The additional parameter does afford greater flexibility: e.g. for small α the sound horizon d s   a 2 /H 0 and thus by fine tuning α<10-5, the data can be perturbatively accommodated
  • 100. Other modifications • The generalized Chaplygin gas in a modified gravity approach, reminiscent of Cardassian models 8 G 1 1 1 ) H  A  ) 3 2 T. Barreiro and A.A. Sen, PRD 70 (2004) • A deformation of the Chaplygin gas – Milne-Born-Infeld theory L A X  bX 2 M. Novello, M. Makler, L.S. Werneck and C.A. Romero, PRD 71 (2005) • Variable Chaplygin gas p ~  an  Zong-Kuan Guo, Yuan-Zhong Zhang, astro-ph/0506091, PLB (2007)
  • 101. 10.4 Tachyon Condensate The failure of the simple Chaplygin gas (CG) does not exhaust all the possibilities for quartessence. The Born-Infeld Lagrangian is a special case of the string-theory inspired tachyon Lagrangian in which the constant A is replaced by a potential L = V () 1  g  ,, . Tachyon models are a particular case of k-essence. It was noted in Zong-Kuan Guo, et al astro-ph/0506091, PLB (2007) that (in the FRW cosmology), the tachyon model is described by the CG equation of state in which the constant A is replaced by a n function of the cosmological scale factor p ~  a  so the model was dubbed “variable Chaplygin gas”.
  • 102. A preliminary analysis of a unifying model based on the tachyon type Lagrangian has been carried out in N.B., G.B. Tupper, R.D. Viollier, PRD 80 (2009) for a potential of the form V ( )  Vn 2 n n  0,1,2 n=0 gives the Dirac-Born-Infeld description of a D-brane - equivalent to the Chaplygin gas It may be shown that the model with n≠0 effectively behaves as a variable Chaplygin gas, with p ~  a 6n  . The much smaller sonic horizon d s ~ a (7 23n ) H 0 enhances condensate formation by 2 orders of magnitude over the simple Chaplygin gas. Hence this type of model may salvage the quartessence scenario.
  • 103. 10.5 Entropy Perturbations One way to deal with the structure formation problem, is to assume entropy perturbations such that the effective speed of sound cs vanishes. In that picture we assume that A is not a constant so we have p = cs2  A/ = 0 even if cs≠0. But in a single field model it is precisely the adiabatic speed of sound that governs the evolution. Hence, entropy perturbations require the introduction of a second field on which A depends.
  • 104. 10.5 Entropy Perturbations One way to deal with the structure formation problem, is to assume entropy perturbations such that the effective speed of sound cs vanishes. This scenario has been suggested as a possible way out of the structure formation problem immanent to all DM/DE unification models. It has been noted by R. Reis et al Phys. Rev. D 69 (2004) that the root of the structure formation problem is the term ∇2δp in perturbation equations, equal to cs2∇2δ for adiabatic perturbations, and if there are entropy perturbations such that δp = 0, no difficulty arises.
  • 105. Suppose that the matter Lagrangian depends on two degrees of freedom, e.g., a Born-Infeld scalar field θ and one additional scalar field φ. In this case, instead of a simple barotropic form p=p(ρ), the equation of state is parametric p=p(θ,φ) ρ=ρ(θ,φ) and involves the entropy density (entropy per particle) s=s(θ,φ) The corresponding perturbations p p  p       s s  s      
  • 106. The speed of sound is the sum of two nonadiabatic terms 1 cs2  p p s  s  p     S 0         Thus, even for a nonzero ∂p/∂ the speed of sound may vanish if the second term on the right-hand side cancels the first one. This cancellation will take place if in the course of an adiabatic expansion, the perturbation δφ grows with a in the same way as δ . In this case, it is only a matter of adjusting the initial conditions of δφ with δ to get cs=0.
  • 107. Obviously, this scenario cannot work for simple or generalized Chaplygin gas models as these models are adiabatic. We have made attempts at realizing the nonadiabatic scenario in some 2 component models (hybrid Chaplygin gas and Kalb- Ramond Chaplygin gas) N. B., G.B. Tupper, and R.D. Viollier, JCAP 0510 (2005); N. B., G.B. Tupper, and R.D. Viollier, J. Phys. A40,(2007), Aside from negating the simplicity of the one-field model, our analysis has convinced us that even if δp=0 is arranged as an initial condition, it is all but impossible to maintain this condition in a realistic model for evolution.
  • 108. 10.6 Dusty Dark Energy Another way to bypass the structure formation problem is to impose a constraint on pressure such that the pressure gradient is parallel to the fluid 4-velocity. The model called Dusty Dark Energy comprises two scalar fields λ and φ, λ being a Lagrange multiplier which enforces a constraint between φ and its kinetic energy term X . Starting from the action  1  S =  d 4 x  g  K  , X )    X  V  ), 2   where K (φ,X) is an arbitrary function of X and φ. The field λ is a Lagrange multiplier and does not have a kinetic term, while X is a standard kinetic term for φ.
  • 109. The energy momentum tensor takes the usual perfect fluid form with the 4-velocity as before u = , / X The λ field equation 1 S 1 = X  V ( ) = 0  g  2 imposes a constraint such that the pressure p = K ( , X ) = K ( ,2V ( )) becomes a function of φ only. Then, its gradient is proportional to φ,μ and hence parallel to the 4-velocity. In this way cs is always identically zero. In particular, cosmological perturbations reproduce the standard hydrodynamic behavior in the limit of vanishing cs . In a certain limit this model reproduces the evolution history of ΛCDM, with some potentially measurable differences.
  • 110. Fate of the universe Solving the mystery of cosmic acceleration will reveal the destiny of our universe. Possible scenarios: 1. If the dark energy density ρΛ is constant or increasing with time, in 100 billion years or so all but a few hundred galaxies will be far too redshifted to be seen. 2. But if the dark energy density decreases and matter becomes dominant again, our cosmic horizon will grow, revealing more of the universe. 3. Even more extreme (and lethal) futures are possible. If dark energy density increases rather than decreases, the universe will eventually undergo a “hyper speedup” that would result in a Big Rip: tearing apart all bound systems, galaxies, solar systems, planets and atomic nuclei, in that order.
  • 111. 11 Conclusions and Outlook  We have discussed a just a few out of many attempts to unify DE/DM. The above mentioned examples illustrate more or less conventional trends in modern cosmology.  There exist various alternative ideas such as modified theories of gravity. One of the popular ideas is the socalled brane world cosmology where our world is a fourdimensional membrane submerged in a five-dimensional space.  Alternative theories explain more or less successfully a part of the phenomena related to dark energy and dark matter, but up until now there is no completely satisfactory theory which would solve all the puzzles.  In any case, there remains a lot of work to be done for theoretical physicists.
  • 113. Thomas-Fermi correspondence Under reasonable assumptions in the cosmological context there exist an equivalence Complex scalar field theories (canonical or phantom) Kinetic k-essence type of models
  • 114. Consider L g    ,  V (|  | m ) * , 2 2 Thomas-Fermi approximation   2 e  im ,  m ,  i, TF Lagrangian where X g  ,,  LTF  4  XY  U (Y ) /m Y 2 2m 2 U (Y )  V (Y ) / m4
  • 115. Equations of motion for φ and θ U X 0 Y  (Yg , );  0 We now define the potential W(X) through a Legendre transformation W ( X )  U (Y )  XY with X  UY and Y  WX U UY  Y W WX  X
  • 116. correspondence Complex scalar FT L g  *,,  V (|  |2    2 Kinetic k-essence FT m2 ) e im Eqs. of motion ( 2 g  , );  0 1 dV g ,,  2 0 2 m d | |  L  m4W ( X ) X  g  , , Eq. of motion (WX g  , );  0 Parametric eq. of state p  m4W   m4 (2 XWX  W )
  • 117. Current conservation Klein-Gordon current   ig  (*,  * ) j , kinetic k-essence current j   2m2WX g  , U(1) symmetry shift symmetry   ei     c
  • 118. Example: Quartic potential Scalar field potential 2 V  0  m0 |  |2  |  |4 V 2 U (Y )  1 1  1 Y  2   2 2  8 Kinetic k-essence 1 1  W(X )   X   2 2  2
  • 119. Example: Chaplygin gas Scalar field potential  |  |2 m2  4 V m  2   2  | |   m Scalar Born-Infeld FT 1  U (Y )   Y   Y  W ( X )  2 1  X