Dark Matter, Dark Energy, and

Unification Models

Neven Bilić

Ruđer Bošković Institute

Zagreb

1 BIG BANG

2 History of the Early Universe - Universe Creation

3 History of the Early Universe – Inflation

4 Cosmological Observations and History of the Universe

5 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 1

9 Dark Energy

9.1 Vacuum Energy – Cosmological Constant

9.2 Quintessence

9.3 Phantom quintessence

9.4 k-essence

Lecture 2

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

Lecture 3

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!

1. Big Bang

C.H. Lineweaver and T. Davis, Misconceptions About the BIG BANG, Scientific

American, March 2005

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 hh – Planck constant

Borrowing a small amount of energy (E0 ) from the

vacuum is allowed for a long time (t )!

As a consequence, the lifetime of the Universe is

almost infinite

2. Early Universe - Creation

3. Early Universe - Inflation

A short period of inflation follows – very rapid

expansion -

1025 times in 10-32 s.

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 horizon problem.

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:

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.

510510= ,v Hd

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

Solar System Foramation, First Supernove

Galaxy and Star Formation

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

Quantum Gravity Era

Universe Today

10-10 s

10-43 s

100 000 years

10-34 s

100 s

1 billion years

13 billion years

5 billion yeras

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

• Cosmic microwave background radiation -

very homogeneous in all directions

• Big Bang nucleosynthesis - proportions of

light elements (H, D, He, Li)

v = Hd

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

a=0.5

a=0.75

a=1

recession

velocity

Hubble’s law –

Universe expansion

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 - Hubble radius

RH

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!

Measuring CMB; the temperature map of the sky.

KT 2.723=

KT 100=D

KT 200=D

Angular (multipole) spectrum of the fluctuations of the CMB

(Planck 2013)

• 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 ...)

6. Theoretical Foundation of Modern

Cosmology

General Relativity – Gravity

g - metric tensor

R - Riemann curvature tensor

- cosmological constant

T - energy – momentum tensor

18

2R g R T g

Matter is described by a perfect fluid

u - fluid velocity

T- energy-momentum tensor

p - pressure

ρ - energy density

( )T p u u pg

Perfect Fluid

0=;T

0=;

Tu

0,=)(3 pH

,1

= ,

ph

pu

.=;=;=3 ;,;

uuuuuH

uugh =

The energy momentum conservation

yields, as its longitudinal part , the continuity equation

and, as its transverse part, the Euler equation

where

projector onto the three-space orthogonal to

H is the local Hubble parameter.

u

Homogeneity and isotropy of space

- cosmological scale

22 2 2 2 2

2( )

1

drds dt a t r d

kr

( )a t

Cosmological principle

the curvature constant k takes on the values 1, 0, or -1,

for a closed, flat, or open universe, respectively.

– cosmological scale

• radiation

• matter

• vacuum

4( 3 )

3

Ga p

; 0T

2

2

8

3

a k G

a a

expansion ratea

Ha

3(1 )wa w p

Various kinds of the cosmic fluids with different w

4

R R 3 1 3p w a 3

M 0 0p w a 01p w a

k 0 open

0 flat

0 closed

Friedman equations

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

From astronomical

observations:

luminous matter (stars,

galaxies, gas ...)

lum/cr0.5%

From the light element

abundances and comparison

with the Big Bang

nucleosynthesis:

barynic matter(protons,

neutrons, nuclei) Bar/cr5%

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%

What does the Universe consist of?

tot

0.05BB

DM

DM

tot

0.26

tot

0.69

DM 1B

tot crit

These fractions change with time but for a

spatialy flat Universe the following always holds:

Density fractions of various kinds of matter today with

respect to the total density

3 4 1/2

0( ) ( )M RH a H a a

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

2 1

0 100Gpc/s (14.5942Gyr) , 0.67H h h

1 1

3 1/2

00 0 0

1

( )

T

M

da da aT dt

aH H a

Age of the Universe is then

13.78GyrT

Age of the Universe

How Large is the Observable Universe?

S The universe is 14 billion years

old, so the radius of the

observable part is 14 billion light-

years. 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

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 ( ) 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.

M1210

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

when even globular cluster masses ( )

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.

M610

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?

9. Dark Energy

Lecture 2

New term: Dark Energy – fluid with negative

presssure - generalization of the concept of vacuum energy

Accelerated expansion 0

One possible explanation is the existence of a fluid

with negative pressure such that

and in the second Friedmann equation the universe

acceleration becomes positive

3 0p

a

cosmological constant vacuum energy density

with equation of state p=-ρ. Its negative pressure may

be responsible for accelerated expansion!

accelerating

time

Accelerating universe?

open

closed

Cosmological scale

today

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.

Time dependence of the DE density

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:

antiparticle

particle

quark

antiquark

e+

W+

W-

The vacuum is not

empty.

e-

Damn! Quantum fluctuations

of the vacuum

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

Doppler redshift

Cosmological redshift is similar to the 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

( )/5 5

L Mpc 10 m MD

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.

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.

Hubble diagram

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)

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

Test of the alternative hypotheses

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

Field theory description of a perfect fluid if X>0

1( )

2X V L

( )T p u u pg

, ,

2

det

ST g

gg

L

P.Ratra, J. Peebles PRD 37 (1988)

4 ( , )S d x X L, ,X g

The coincidence problem is somewhat ameliorated in

quintessence – a canonical scalar field with selfinteraction

effectively providing a slow roll inflation today

where

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

1( )

2p X V L

,u

X

1( )

2X X V L

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( )

2p X V L

1( )

2X V

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

),(

16= 4 X

G

RgxdS

L

where L is the most general Lagrangian, which depends

on a single scalar field of dimension , and on the

dimensionless quantity For X>0 , the energy

momentum tensor takes the perfect fluid form, ,X g

,)(=2= ,, gpuupgT X LL

XX /LLwhere

Kinetic k-essence The Lagrangian is a function of X

only. In this case

Examples

)(= Xp L )()(2= XXX X LL

The associated hydrodynamic quantities are

),(= Xp L ).,(),(2= XXX X LL

To this class belong the ghost condensate

BXAX 2)(1=)(L

and the scalar Born-Infeld model

XAX 1=)(L

L(X) L(X)

X X

N. Arkani-Hamed et al , JHEP 05 (2004)

R.J. Scherrer, PRL 93 (2004)

or

Ghost condensate model

de-Sitterde-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

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)

Ap

In a homogeneous model the conservation equation yields the

Chaplygin gas density as a function of the scale factor a

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

6BA

a

Scalar Born-Infeld theory

BI 1A X L

BI, , BI2T g

X

L L

( )T p u u pg ,u

X

with

, ,X g

perfect fluid

Chaplygin gas has an equivalent scalar field formulation.

Considering a kinetic k-essence type of Lagrangian

the induced metric (“pull back”) of the bulk metric Gab

x

X

x

XGg

ba

ab

ind

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

1 ind

DBI ( 1) det( )p pS A d x g

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

Xa

xi

x0θ4 44

for 0...3

0 1

G g

G G

We find a k-essence type of theory

with

and hence

4 2

DBI 1S A dx X , ,X g

2

2; 1

1

Ap A X

X

Ap

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 α =0

Power spectrum for p=-A/ρα for various α

H.B. Sandvik et al, PRD 2004

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

is small until a ~ 1, the accumulated comovingsize of the acoustic horizon

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.

2

s 2

S

p Ac

1 7 / 2ss 0

cd dt H a

a

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

2 2 ,

;

3 3 4 ( 3 ) =h p

G pp

H H

2 =

( ; )= h h u h

H

We thus obtain an evolution equation for H . Owing to the

definition of the four-velocity and orthogonality

we may write 0=

uh

.= ,

2

,

hcph 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

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 = ( ) /

2 23 3 4 ( 3 ) = 0G p H H

H H = H

2 22

2

3 4 12 (1 ) = 0

2 31 1

s

i i

c

a x x

H H

In the Newtonian approximation subtracting the

background, eliminating δH and ∂tδH and neglecting

shear we find

The root of the structure formation problem is the last

term, as may be understood if we solve the evolution

equation at linear order

with the solution (in k-space)

• 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.

22 s

2

32

2

cH H

a D 0

1/ 4

5/ 4 s

s

ss2

( )

for 1

cos ( )for 1

k

k

k

a J d k

a d k

d kd k

a

behaving asymptotically as

k – comoving wave number

ds –comoving size of the acoustic

horizon

In the case of the Chaplygin gas we have

0

7/2/Hads

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.

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.

Nonlinear evolution of inhomogeneities

gas + condensate

collapse during

expansion

nearly homogeneous gas

at dec 1089z

Two phase structure – mixture of CDM in the form of

condensate and DE in the form of gas

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

γ is the determinant of γij in comoving gauge.

N.B., G.B. Tupper, R.D. Viollier, PLB 535 (2002)

BA

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)

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

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)

2 22

2

3 4 12 (1 ) = 0

2 31 1

s

i i

c

a x x

H H

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

10,=

Ap

The additional parameter does afford greater flexibility: e.g.

for small α the sound horizon and thus by fine

tuning α<10-5, the data can be perturbatively accommodated0

2/Hads

Other modifications

• The generalized Chaplygin gas in a modified gravity approach, reminiscent of Cardassian models

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

• A deformation of the Chaplygin gas – Milne-Born-Infeldtheory

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

• Variable Chaplygin gas

2A X bX L

) )1 1

2 18

3

GH A

Zong-Kuan Guo, Yuan-Zhong Zhang, astro-ph/0506091, PLB (2007)

~ np a

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

, ,= ( ) 1 .V g

L

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

function of the cosmological scale factor so the

model was dubbed “variable Chaplygin gas”.

~ np a

A preliminary analysis of a unifying model based on the

tachyon type Lagrangian has been carried out in

for a potential of the form

N.B., G.B. Tupper, R.D. Viollier, PRD 80 (2009)

2( ) 0,1,2n

nV V n

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 . The much

smaller sonic horizon enhances condensate

formation by 2 orders of magnitude over the simple Chaplygin

gas. Hence this type of model may salvage the quartessencescenario.

(7 2 3 )

0~ n

sd a H

6~ np a

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

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.

0=/= 2 Acp s

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.

10.5 Entropy Perturbations

Suppose that the matter Lagrangian depends on two degrees of freedom, e.g., a Born-Infeld scalar field θand 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 pp

s ss

The speed of sound is the sum of two nonadiabatic

terms

Thus, even for a nonzero ∂p/∂ the speed of soundmay 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.

1

2

s

0S

p p s s pc

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)

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.

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),

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

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.

) ) ,2

1,= 4

VXXKgxdS

The energy momentum tensor takes the usual perfect fluid

form with the 4-velocity as before ,= /u X

The λ field equation

imposes a constraint such that the pressure

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.

0=)(2

1=

1

VX

S

g

))(,2(=),(= VKXKp

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 so-

called brane world cosmology where our world is a four-

dimensional 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.

Thomas-Fermi correspondence

Complex scalar field

theories (canonical or

phantom)

Kinetic k-essence type

of models

Under reasonable assumptions in the cosmological

context there exist an equivalence

Consider

Thomas-Fermi approximation

TF Lagrangian

, m

, ,i

4

TF / ( )m XY U Y L

, ,X g

2

22Y

m

4( ) ( ) /U Y V Y m

* 2 2

, , (| | )g V m

L2

ime

where

Equations of motion for φ and θ

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

transformation

, ;( ) 0Yg

0U

XY

Y

UU

Y

( ) ( )W X U Y XY

YX U XY Wwith andX

WW

X

correspondence

Complex scalar FT

Eqs. of motion

Kinetic k-essence FT

Eq. of motion

Parametric eq. of state

* 2 2

, , (| | )g V m

L

2

ime

2

, ;( ) 0g

, , 2 2

10

| |

dVg

m d

4 ( )m W XL

, ,X g

, ;( ) 0XW g

4 4(2 )Xp m W m XW W

Current conservation

Klein-Gordon current

U(1) symmetry

kinetic k-essence current

shift symmetry

* *

, ,( )j ig

2

,2 Xj m W g

ie c

Example: Quartic potential

Scalar field potential

Kinetic k-essence2 2 4

0 0 | | | |V V m

2

2

1 1 1( )

2 2 8U Y Y

21 1

( )2 2

W X X

Example: Chaplygin gas

Scalar field potential

Scalar Born-Infeld FT

2 24

2 2

| |

| |

mV m

m

( ) 2 1W X X 1

( )U Y YY